CASIO BASIC (CASIO fx-9750giii), \$(a/b)+ab\$ additions, 40 bytes

?→A
?→B
While D<A
Isz C
For 0→E~D To B
C+D→D
Next
WhileEnd
C

translation of my answer in jBasher2

jBasher2, \$(a/b)+ab \$ additions, 333 bytes

(where a is the dividend and b is the divisor)

create a with type number
create b with type number
create c with type number
create d with type number
create e with type number
ask for input
set that to a
ask for input
set that to b
while d < a
add 1 by c
set that to c
set 0 to e
set 0 to d
while e < b
add c by d
set that to d
add 1 by e
set that to e
endwhile
endwhile
output c

pretty self explanatory

Tcl, 96 bytes, a additions

where a is the divisor.

Non-competing, but I just wanted to have fun.

proc M f\ g {
	time {incr p $f} $g
	set p
}

proc D p\ f {
	while {[M [incr i] $f]<$p} {}
set i}

Try it online!

J, 0 additions, 14 bytes

Inspired by Alexei Kopylov's answer.

f=:[{]#i.@>:@[

Uses no maths at all:

f=:                    NB. define function f
             [         NB. take left argument,
          >:@          NB. increment it,
       i.@             NB. generate the list [0..left arg+1)
     ]#                NB. replicate each item in the list by the right argument
                       NB. (so if ]=2, list becomes 0 0 1 1 2 2 3 3 ...)
   [{                  NB. select the ['th item from that list.

C — 85591 Additions

Here we go. I think this might be optimal. It uses a technique of "reverse division" whereby through long multiplication it builds up the largest number q such that q * b <= a, using only + and <=. It is very, very fast.

#include <stdio.h>
#include <assert.h>

// Division function.
q,u,v,z=0;s(a,b){return!a?b:!b?a:(z++,a+b);}
d(a,b,p){if((v=s(b,b))<=a)d(a,v,s(p,p));if((v=s(u,b))<=a)u=v,q=s(p,q);}
divide(a,b){u=q=0;d(a,b,1);return q;}

// Test driver.
main(){for(int a=1;a<=200;a++)for(int b=1;b<=a;b++)assert(divide(a,b)==q);
printf("%d additions\n",z);}

Notes:

1. s(a,b) returns the sum a+b and increments counter variable z each time an addition is performed. If either a or b is zero, the unnecessary addition is avoided.
2. d(a,b,p) is a recursive function to build up the internal portions for comparison and addition. It uses global variables q, u, and v. Maximum recursion depth is the number of bits in a, and the recursion is linear rather than a tree. (Note: the b in this function is the original b multiplied by a power of 2.)
3. divide(a,b) returns floor(a/b) as required.
4. Compiles with warnings (because code is golfed). Runs fine.

Haskell 0 additions, 29 bytes

n/m=[i|i<-[0..],_<-[1..m]]!!n

this redefines the division operator (/). it works by making a list of 0 to infinity where each item is repeated m times, and then choosing the nth element of the list (using a 0-based index).

Java: 92 987 additions

I use binary recursion, that a/b == 2 * a/(2b) + maybe 1. For that divisor and remainder are needed. There would normally be a subtraction a % (2b) - b, but that is resolved by holding the remainder as (rem, remNegative). And 2b = b+b of course.

static int add_used;

static int add(int a, int b) {
    if (a == 0)
        return b;
    if (b == 0)
        return a;
    ++add_used;
    return a + b;
}

private static class DivRem {
    int div;
    int rem;
    int remNegative;

    DivRem(int div, int rem) {
        this.div = div;
        this.rem = rem;
    }
}

public static int divide(int a, int b) {
    add_used = 0;
    return divrem(a, b).div;
}

public static DivRem divrem(int a, int b) {
    if (b > a) {
        return new DivRem(0, a);
    }
    DivRem dr = divrem(a, add(b, b));
    dr.div = add(dr.div, dr.div);
    if (dr.rem >= add(b, dr.remNegative)) {
        dr.div = add(dr.div, 1);
        //dr.rem = add(dr.rem,  -b);
        dr.remNegative = add(dr.remNegative,  b);
    }
    return dr;
}

private static void test(int a, int b) {
    boolean okay = a/b == divide(a, b);
    System.out.printf("%d / %d = %d :: %d : #%d  %s%n", a, b, a/b,
        divide(a, b), add_used, okay);
}

public static void main(String[] args) {
   //test(2352, 324);
   int n = 0;
   for (int a = 1; a <= 200; ++a) {
       for (int b = 1; b <= a; ++b) {
           //test(a, b);
           divide(a, b);
           n += add_used;
       }
   }
   System.out.println("Additions: " + n);
}

In Ruby,

def divide(a,b)
  n, d = 'x' * a, 'x' * b
  l = []
  (l << 'x'; d << 'x' * b) while n.size >= d.size
  l.size
end  

I don't know TCL, but I suspect this is the same approach as @Johannes ' (first) answer.

My solution is C/C++ code and it makes many additions (200402), but anyway...

#include <iostream>

int total = 0;

int sum(int a, int b)
{
    ++total;
    return a + b;
}

int divide(int a, int b)
{
    int x = 1;
    if (a < b)
        return 0;
    else
        return sum(x, divide(sum(a, -b), b));
}

int main()
{
    for (int i = 1; i <= 200; ++i)
        for (int j = 1; j <= 200; ++j)
        {
            if (divide(i, j) != (i / j))
                std::cout << "Failure: a=" << i << " b=" << j << "\n";
        }

    std::cout << "Total additions: " << total << std::endl;
    system("pause");
}

And the output is:

Total additions: 200402
Press any key to continue . . .

C# - 0 additions

using System.Collections.Generic;
using System.Linq;

static int Divide(int a, int b)
{
    var ints = new List<int>(a);
    while (ints.Count < a)
        ints.AddRange(Enumerable.Range(1, b));

    return ints.Select((x, i) => x == b && i < a).Count(x => x);
}

Populates a list of integers with 1..b repeated a times. The number of times b appears (except for the occurrence with an index > a) is the result.

I'm not sure if the list is allowed by the rules, but I'm submitting this in the spirit of the other posts which aren't taking the rules all that seriously (after all, not using addition at all is basically bypassing the challenge altogether).

Python - 0 additions

from itertools import repeat, count

def divide(a, b):
    i = repeat(0, a)
    try:
        for j in count():
            for k in repeat(0, b):
                next(i)
    except:
        return j

This uses an iterator of length a, and consumes it in groups of b until StopIteration is raised. At this point j contains the result.

R - 0 addition

divide<-function(a,b){
    options(warn=-1)
    A<-matrix(1:b,nrow=a,ncol=1)
    length(split(A,A)[[b]])
    }

Uses R vector recycling.
Second line creates a matrix of length a populated by a vector of length bwhich is recycled until reaching length a.
Third line split the matrix according to its value and return the length of the last element (hence the result of the integer division of a by b).
Populating a matrix with a vector which length is not a multiple of the length of the matrix throws a warning but if we suppress warning beforehand (line 1) it works.

To give a concrete example if we divide 5 by 3, A will be a vector containing 1 2 3 1 2 (i. e. 1 2 3 recycled to a length 5). The result of the splitting operation will be a list with the first element containing 1 1, the second 2 2 and the third 3 (since there is only one 3 in A). The result is therefore 1.

Mathematica 100201 additions

This adds the divisor, b, to c (which is initialized at 0) as long as the running total is less than or equal to the dividend, a. It also appends the current value of c to a list, t, without performing any arithmetic operation.

When the While loop terminates the function outputs the length of t, which will correspond exactly to the quotient of integer division. Thus the number of additions for any given divide[a,b] will equal precisely the quotient.

100201 is the sum of the quotients in the 200 by 200 table. That's how many times c was incremented by b. No other additions were required. Only positive integers were used.

divide[a_, b_] := Module[{c = 0, t = {}}, While[c <= a, t = Append[t, c]; c += b]; 
Length[Rest@t]]

It's more efficient to make a lookup table, after which each search will be almost instantaneous.

n = 200;
d[a_, b_] := Module[{c = 0, t = {}}, While[c <= a, t = Append[t, c]; c += b]; 
Length[Rest@t]]
quotients = PadRight[#, n] & /@ Table[d[i, j], {i, 1, n}, {j, 1, i}];
divide[a_, b_] := quotients[[a, b]]

Usage

divide[97, 13]

7

Use this implementation in java, 199206 additions

public int divide(int a, int b){
    int counter = 0;
    int c = 0;
    if(b==1){
        return a;
    }
    if(a==b){
        return 1;
    }
    else{
        boolean done = false;
        while(!done){
            c = add(c, b);
            if(a<c){
                done = true;
            }
            counter = add(counter,1);
        }
        return counter;

    }
}

Following are the helper functions

public static void main(String[] args) {
    Main main = new Main();

    for(int a = 1; a<=200; a++){    
        for(int b=1;b<=a;b++){
            main.divide(a, b);
        }
    }

    System.out.println("Number of additions: "+numberOfAdds);
}

public int add(int a, int b){
    numberOfAdds++;
    return (a+b);
}

//a lies between 1 and 200, b lies between 1 and a.

int divide(int a,int b){
int x=a,y=b;
int count=1;
while(y<x){
y+=y;
count++;
}
return count;
}

C++ ,100201

for(int a = 1; a<=200; a++){    
        for(int b=1;b<=a;b++){
    iter1 = iter2 = b; 
cout<<a<<" "<<b<<endl;   

c1 =0;
while(iter1 <= a)
{
    iter1 = iter1 + iter2;
    c1 ++;
    nadd++;
}
cout<<"Quotient : "<<c1;
cout<<" Remainder :"<<a - (iter1 - iter2)<<endl;    
}  
}
cout<<"total of add "<<nadd;

Writing rules is hard, these rules in particular contain incentive to avoid additions at all costs.

Is there a prize for the most ridiculous answer?

JavaScript - 0 additions

Now with fallback method that does a hulking solution for larger a's and b's, and a slightly more compact structure in order not to bust the character limit. (Pfff, 30000 characters. What is this? Twitter?) Still no additions in the measured scope.

function divide(a,b){
    if(a<b){
        return 0
    }
    if(b==1){
        return a
    }
    if(b==2){
        if(a<4){return 1}
        if(a<6){return 2}
        if(a<8){return 3}
        if(a<10){return 4}
        if(a<12){return 5}
        if(a<14){return 6}
        if(a<16){return 7}
        if(a<18){return 8}
        if(a<20){return 9}
        if(a<22){return 10}
        if(a<24){return 11}
        if(a<26){return 12}
        if(a<28){return 13}
        if(a<30){return 14}
        if(a<32){return 15}
        if(a<34){return 16}
        if(a<36){return 17}
        if(a<38){return 18}
        if(a<40){return 19}
        if(a<42){return 20}
        if(a<44){return 21}
        if(a<46){return 22}
        if(a<48){return 23}
        if(a<50){return 24}
        if(a<52){return 25}
        if(a<54){return 26}
        if(a<56){return 27}
        if(a<58){return 28}
        if(a<60){return 29}
        if(a<62){return 30}
        if(a<64){return 31}
        if(a<66){return 32}
        if(a<68){return 33}
        if(a<70){return 34}
        if(a<72){return 35}
        if(a<74){return 36}
        if(a<76){return 37}
        if(a<78){return 38}
        if(a<80){return 39}
        if(a<82){return 40}
        if(a<84){return 41}
        if(a<86){return 42}
        if(a<88){return 43}
        if(a<90){return 44}
        if(a<92){return 45}
        if(a<94){return 46}
        if(a<96){return 47}
        if(a<98){return 48}
        if(a<100){return 49}
        if(a<102){return 50}
        if(a<104){return 51}
        if(a<106){return 52}
        if(a<108){return 53}
        if(a<110){return 54}
        if(a<112){return 55}
        if(a<114){return 56}
        if(a<116){return 57}
        if(a<118){return 58}
        if(a<120){return 59}
        if(a<122){return 60}
        if(a<124){return 61}
        if(a<126){return 62}
        if(a<128){return 63}
        if(a<130){return 64}
        if(a<132){return 65}
        if(a<134){return 66}
        if(a<136){return 67}
        if(a<138){return 68}
        if(a<140){return 69}
        if(a<142){return 70}
        if(a<144){return 71}
        if(a<146){return 72}
        if(a<148){return 73}
        if(a<150){return 74}
        if(a<152){return 75}
        if(a<154){return 76}
        if(a<156){return 77}
        if(a<158){return 78}
        if(a<160){return 79}
        if(a<162){return 80}
        if(a<164){return 81}
        if(a<166){return 82}
        if(a<168){return 83}
        if(a<170){return 84}
        if(a<172){return 85}
        if(a<174){return 86}
        if(a<176){return 87}
        if(a<178){return 88}
        if(a<180){return 89}
        if(a<182){return 90}
        if(a<184){return 91}
        if(a<186){return 92}
        if(a<188){return 93}
        if(a<190){return 94}
        if(a<192){return 95}
        if(a<194){return 96}
        if(a<196){return 97}
        if(a<198){return 98}
        if(a<200){return 99}
        if(a<202){return 100}
    }
    if(b==3){
        if(a<6){return 1}
        if(a<9){return 2}
        if(a<12){return 3}
        if(a<15){return 4}
        if(a<18){return 5}
        if(a<21){return 6}
        if(a<24){return 7}
        if(a<27){return 8}
        if(a<30){return 9}
        if(a<33){return 10}
        if(a<36){return 11}
        if(a<39){return 12}
        if(a<42){return 13}
        if(a<45){return 14}
        if(a<48){return 15}
        if(a<51){return 16}
        if(a<54){return 17}
        if(a<57){return 18}
        if(a<60){return 19}
        if(a<63){return 20}
        if(a<66){return 21}
        if(a<69){return 22}
        if(a<72){return 23}
        if(a<75){return 24}
        if(a<78){return 25}
        if(a<81){return 26}
        if(a<84){return 27}
        if(a<87){return 28}
        if(a<90){return 29}
        if(a<93){return 30}
        if(a<96){return 31}
        if(a<99){return 32}
        if(a<102){return 33}
        if(a<105){return 34}
        if(a<108){return 35}
        if(a<111){return 36}
        if(a<114){return 37}
        if(a<117){return 38}
        if(a<120){return 39}
        if(a<123){return 40}
        if(a<126){return 41}
        if(a<129){return 42}
        if(a<132){return 43}
        if(a<135){return 44}
        if(a<138){return 45}
        if(a<141){return 46}
        if(a<144){return 47}
        if(a<147){return 48}
        if(a<150){return 49}
        if(a<153){return 50}
        if(a<156){return 51}
        if(a<159){return 52}
        if(a<162){return 53}
        if(a<165){return 54}
        if(a<168){return 55}
        if(a<171){return 56}
        if(a<174){return 57}
        if(a<177){return 58}
        if(a<180){return 59}
        if(a<183){return 60}
        if(a<186){return 61}
        if(a<189){return 62}
        if(a<192){return 63}
        if(a<195){return 64}
        if(a<198){return 65}
        if(a<201){return 66}
    }
    if(b==4){
        if(a<8){return 1}
        if(a<12){return 2}
        if(a<16){return 3}
        if(a<20){return 4}
        if(a<24){return 5}
        if(a<28){return 6}
        if(a<32){return 7}
        if(a<36){return 8}
        if(a<40){return 9}
        if(a<44){return 10}
        if(a<48){return 11}
        if(a<52){return 12}
        if(a<56){return 13}
        if(a<60){return 14}
        if(a<64){return 15}
        if(a<68){return 16}
        if(a<72){return 17}
        if(a<76){return 18}
        if(a<80){return 19}
        if(a<84){return 20}
        if(a<88){return 21}
        if(a<92){return 22}
        if(a<96){return 23}
        if(a<100){return 24}
        if(a<104){return 25}
        if(a<108){return 26}
        if(a<112){return 27}
        if(a<116){return 28}
        if(a<120){return 29}
        if(a<124){return 30}
        if(a<128){return 31}
        if(a<132){return 32}
        if(a<136){return 33}
        if(a<140){return 34}
        if(a<144){return 35}
        if(a<148){return 36}
        if(a<152){return 37}
        if(a<156){return 38}
        if(a<160){return 39}
        if(a<164){return 40}
        if(a<168){return 41}
        if(a<172){return 42}
        if(a<176){return 43}
        if(a<180){return 44}
        if(a<184){return 45}
        if(a<188){return 46}
        if(a<192){return 47}
        if(a<196){return 48}
        if(a<200){return 49}
        if(a<204){return 50}
    }
    if(b==5){
        if(a<10){return 1}
        if(a<15){return 2}
        if(a<20){return 3}
        if(a<25){return 4}
        if(a<30){return 5}
        if(a<35){return 6}
        if(a<40){return 7}
        if(a<45){return 8}
        if(a<50){return 9}
        if(a<55){return 10}
        if(a<60){return 11}
        if(a<65){return 12}
        if(a<70){return 13}
        if(a<75){return 14}
        if(a<80){return 15}
        if(a<85){return 16}
        if(a<90){return 17}
        if(a<95){return 18}
        if(a<100){return 19}
        if(a<105){return 20}
        if(a<110){return 21}
        if(a<115){return 22}
        if(a<120){return 23}
        if(a<125){return 24}
        if(a<130){return 25}
        if(a<135){return 26}
        if(a<140){return 27}
        if(a<145){return 28}
        if(a<150){return 29}
        if(a<155){return 30}
        if(a<160){return 31}
        if(a<165){return 32}
        if(a<170){return 33}
        if(a<175){return 34}
        if(a<180){return 35}
        if(a<185){return 36}
        if(a<190){return 37}
        if(a<195){return 38}
        if(a<200){return 39}
        if(a<205){return 40}
    }
    if(b==6){
        if(a<12){return 1}
        if(a<18){return 2}
        if(a<24){return 3}
        if(a<30){return 4}
        if(a<36){return 5}
        if(a<42){return 6}
        if(a<48){return 7}
        if(a<54){return 8}
        if(a<60){return 9}
        if(a<66){return 10}
        if(a<72){return 11}
        if(a<78){return 12}
        if(a<84){return 13}
        if(a<90){return 14}
        if(a<96){return 15}
        if(a<102){return 16}
        if(a<108){return 17}
        if(a<114){return 18}
        if(a<120){return 19}
        if(a<126){return 20}
        if(a<132){return 21}
        if(a<138){return 22}
        if(a<144){return 23}
        if(a<150){return 24}
        if(a<156){return 25}
        if(a<162){return 26}
        if(a<168){return 27}
        if(a<174){return 28}
        if(a<180){return 29}
        if(a<186){return 30}
        if(a<192){return 31}
        if(a<198){return 32}
        if(a<204){return 33}
    }
    if(b==7){
        if(a<14){return 1}
        if(a<21){return 2}
        if(a<28){return 3}
        if(a<35){return 4}
        if(a<42){return 5}
        if(a<49){return 6}
        if(a<56){return 7}
        if(a<63){return 8}
        if(a<70){return 9}
        if(a<77){return 10}
        if(a<84){return 11}
        if(a<91){return 12}
        if(a<98){return 13}
        if(a<105){return 14}
        if(a<112){return 15}
        if(a<119){return 16}
        if(a<126){return 17}
        if(a<133){return 18}
        if(a<140){return 19}
        if(a<147){return 20}
        if(a<154){return 21}
        if(a<161){return 22}
        if(a<168){return 23}
        if(a<175){return 24}
        if(a<182){return 25}
        if(a<189){return 26}
        if(a<196){return 27}
        if(a<203){return 28}
    }
    if(b==8){
        if(a<16){return 1}
        if(a<24){return 2}
        if(a<32){return 3}
        if(a<40){return 4}
        if(a<48){return 5}
        if(a<56){return 6}
        if(a<64){return 7}
        if(a<72){return 8}
        if(a<80){return 9}
        if(a<88){return 10}
        if(a<96){return 11}
        if(a<104){return 12}
        if(a<112){return 13}
        if(a<120){return 14}
        if(a<128){return 15}
        if(a<136){return 16}
        if(a<144){return 17}
        if(a<152){return 18}
        if(a<160){return 19}
        if(a<168){return 20}
        if(a<176){return 21}
        if(a<184){return 22}
        if(a<192){return 23}
        if(a<200){return 24}
        if(a<208){return 25}
    }
    if(b==9){
        if(a<18){return 1}
        if(a<27){return 2}
        if(a<36){return 3}
        if(a<45){return 4}
        if(a<54){return 5}
        if(a<63){return 6}
        if(a<72){return 7}
        if(a<81){return 8}
        if(a<90){return 9}
        if(a<99){return 10}
        if(a<108){return 11}
        if(a<117){return 12}
        if(a<126){return 13}
        if(a<135){return 14}
        if(a<144){return 15}
        if(a<153){return 16}
        if(a<162){return 17}
        if(a<171){return 18}
        if(a<180){return 19}
        if(a<189){return 20}
        if(a<198){return 21}
        if(a<207){return 22}
    }
    if(b==10){
        if(a<20){return 1}
        if(a<30){return 2}
        if(a<40){return 3}
        if(a<50){return 4}
        if(a<60){return 5}
        if(a<70){return 6}
        if(a<80){return 7}
        if(a<90){return 8}
        if(a<100){return 9}
        if(a<110){return 10}
        if(a<120){return 11}
        if(a<130){return 12}
        if(a<140){return 13}
        if(a<150){return 14}
        if(a<160){return 15}
        if(a<170){return 16}
        if(a<180){return 17}
        if(a<190){return 18}
        if(a<200){return 19}
        if(a<210){return 20}
    }
    if(b==11){
        if(a<22){return 1}
        if(a<33){return 2}
        if(a<44){return 3}
        if(a<55){return 4}
        if(a<66){return 5}
        if(a<77){return 6}
        if(a<88){return 7}
        if(a<99){return 8}
        if(a<110){return 9}
        if(a<121){return 10}
        if(a<132){return 11}
        if(a<143){return 12}
        if(a<154){return 13}
        if(a<165){return 14}
        if(a<176){return 15}
        if(a<187){return 16}
        if(a<198){return 17}
        if(a<209){return 18}
    }
    if(b==12){
        if(a<24){return 1}
        if(a<36){return 2}
        if(a<48){return 3}
        if(a<60){return 4}
        if(a<72){return 5}
        if(a<84){return 6}
        if(a<96){return 7}
        if(a<108){return 8}
        if(a<120){return 9}
        if(a<132){return 10}
        if(a<144){return 11}
        if(a<156){return 12}
        if(a<168){return 13}
        if(a<180){return 14}
        if(a<192){return 15}
        if(a<204){return 16}
    }
    if(b==13){
        if(a<26){return 1}
        if(a<39){return 2}
        if(a<52){return 3}
        if(a<65){return 4}
        if(a<78){return 5}
        if(a<91){return 6}
        if(a<104){return 7}
        if(a<117){return 8}
        if(a<130){return 9}
        if(a<143){return 10}
        if(a<156){return 11}
        if(a<169){return 12}
        if(a<182){return 13}
        if(a<195){return 14}
        if(a<208){return 15}
    }
    if(b==14){
        if(a<28){return 1}
        if(a<42){return 2}
        if(a<56){return 3}
        if(a<70){return 4}
        if(a<84){return 5}
        if(a<98){return 6}
        if(a<112){return 7}
        if(a<126){return 8}
        if(a<140){return 9}
        if(a<154){return 10}
        if(a<168){return 11}
        if(a<182){return 12}
        if(a<196){return 13}
        if(a<210){return 14}
    }
    if(b==15){
        if(a<30){return 1}
        if(a<45){return 2}
        if(a<60){return 3}
        if(a<75){return 4}
        if(a<90){return 5}
        if(a<105){return 6}
        if(a<120){return 7}
        if(a<135){return 8}
        if(a<150){return 9}
        if(a<165){return 10}
        if(a<180){return 11}
        if(a<195){return 12}
        if(a<210){return 13}
    }
    if(b==16){
        if(a<32){return 1}
        if(a<48){return 2}
        if(a<64){return 3}
        if(a<80){return 4}
        if(a<96){return 5}
        if(a<112){return 6}
        if(a<128){return 7}
        if(a<144){return 8}
        if(a<160){return 9}
        if(a<176){return 10}
        if(a<192){return 11}
        if(a<208){return 12}
    }
    if(b==17){
        if(a<34){return 1}
        if(a<51){return 2}
        if(a<68){return 3}
        if(a<85){return 4}
        if(a<102){return 5}
        if(a<119){return 6}
        if(a<136){return 7}
        if(a<153){return 8}
        if(a<170){return 9}
        if(a<187){return 10}
        if(a<204){return 11}
    }
    if(b==18){
        if(a<36){return 1}
        if(a<54){return 2}
        if(a<72){return 3}
        if(a<90){return 4}
        if(a<108){return 5}
        if(a<126){return 6}
        if(a<144){return 7}
        if(a<162){return 8}
        if(a<180){return 9}
        if(a<198){return 10}
        if(a<216){return 11}
    }
    if(b==19){
        if(a<38){return 1}
        if(a<57){return 2}
        if(a<76){return 3}
        if(a<95){return 4}
        if(a<114){return 5}
        if(a<133){return 6}
        if(a<152){return 7}
        if(a<171){return 8}
        if(a<190){return 9}
        if(a<209){return 10}
    }
    if(b==20){
        if(a<40){return 1}
        if(a<60){return 2}
        if(a<80){return 3}
        if(a<100){return 4}
        if(a<120){return 5}
        if(a<140){return 6}
        if(a<160){return 7}
        if(a<180){return 8}
        if(a<200){return 9}
        if(a<220){return 10}
    }
    if(b==21){
        if(a<42){return 1}
        if(a<63){return 2}
        if(a<84){return 3}
        if(a<105){return 4}
        if(a<126){return 5}
        if(a<147){return 6}
        if(a<168){return 7}
        if(a<189){return 8}
        if(a<210){return 9}
    }
    if(b==22){
        if(a<44){return 1}
        if(a<66){return 2}
        if(a<88){return 3}
        if(a<110){return 4}
        if(a<132){return 5}
        if(a<154){return 6}
        if(a<176){return 7}
        if(a<198){return 8}
        if(a<220){return 9}
    }
    if(b==23){
        if(a<46){return 1}
        if(a<69){return 2}
        if(a<92){return 3}
        if(a<115){return 4}
        if(a<138){return 5}
        if(a<161){return 6}
        if(a<184){return 7}
        if(a<207){return 8}
    }
    if(b==24){
        if(a<48){return 1}
        if(a<72){return 2}
        if(a<96){return 3}
        if(a<120){return 4}
        if(a<144){return 5}
        if(a<168){return 6}
        if(a<192){return 7}
        if(a<216){return 8}
    }
    if(b==25){
        if(a<50){return 1}
        if(a<75){return 2}
        if(a<100){return 3}
        if(a<125){return 4}
        if(a<150){return 5}
        if(a<175){return 6}
        if(a<200){return 7}
        if(a<225){return 8}
    }
    if(b==26){
        if(a<52){return 1}
        if(a<78){return 2}
        if(a<104){return 3}
        if(a<130){return 4}
        if(a<156){return 5}
        if(a<182){return 6}
        if(a<208){return 7}
    }
    if(b==27){
        if(a<54){return 1}
        if(a<81){return 2}
        if(a<108){return 3}
        if(a<135){return 4}
        if(a<162){return 5}
        if(a<189){return 6}
        if(a<216){return 7}
    }
    if(b==28){
        if(a<56){return 1}
        if(a<84){return 2}
        if(a<112){return 3}
        if(a<140){return 4}
        if(a<168){return 5}
        if(a<196){return 6}
        if(a<224){return 7}
    }
    if(b==29){
        if(a<58){return 1}
        if(a<87){return 2}
        if(a<116){return 3}
        if(a<145){return 4}
        if(a<174){return 5}
        if(a<203){return 6}
    }
    if(b==30){
        if(a<60){return 1}
        if(a<90){return 2}
        if(a<120){return 3}
        if(a<150){return 4}
        if(a<180){return 5}
        if(a<210){return 6}
    }
    if(b==31){
        if(a<62){return 1}
        if(a<93){return 2}
        if(a<124){return 3}
        if(a<155){return 4}
        if(a<186){return 5}
        if(a<217){return 6}
    }
    if(b==32){
        if(a<64){return 1}
        if(a<96){return 2}
        if(a<128){return 3}
        if(a<160){return 4}
        if(a<192){return 5}
        if(a<224){return 6}
    }
    if(b==33){
        if(a<66){return 1}
        if(a<99){return 2}
        if(a<132){return 3}
        if(a<165){return 4}
        if(a<198){return 5}
        if(a<231){return 6}
    }
    if(b==34){
        if(a<68){return 1}
        if(a<102){return 2}
        if(a<136){return 3}
        if(a<170){return 4}
        if(a<204){return 5}
    }
    if(b==35){
        if(a<70){return 1}
        if(a<105){return 2}
        if(a<140){return 3}
        if(a<175){return 4}
        if(a<210){return 5}
    }
    if(b==36){
        if(a<72){return 1}
        if(a<108){return 2}
        if(a<144){return 3}
        if(a<180){return 4}
        if(a<216){return 5}
    }
    if(b==37){
        if(a<74){return 1}
        if(a<111){return 2}
        if(a<148){return 3}
        if(a<185){return 4}
        if(a<222){return 5}
    }
    if(b==38){
        if(a<76){return 1}
        if(a<114){return 2}
        if(a<152){return 3}
        if(a<190){return 4}
        if(a<228){return 5}
    }
    if(b==39){
        if(a<78){return 1}
        if(a<117){return 2}
        if(a<156){return 3}
        if(a<195){return 4}
        if(a<234){return 5}
    }
    if(b==40){
        if(a<80){return 1}
        if(a<120){return 2}
        if(a<160){return 3}
        if(a<200){return 4}
        if(a<240){return 5}
    }
    if(b==41){
        if(a<82){return 1}
        if(a<123){return 2}
        if(a<164){return 3}
        if(a<205){return 4}
    }
    if(b==42){
        if(a<84){return 1}
        if(a<126){return 2}
        if(a<168){return 3}
        if(a<210){return 4}
    }
    if(b==43){
        if(a<86){return 1}
        if(a<129){return 2}
        if(a<172){return 3}
        if(a<215){return 4}
    }
    if(b==44){
        if(a<88){return 1}
        if(a<132){return 2}
        if(a<176){return 3}
        if(a<220){return 4}
    }
    if(b==45){
        if(a<90){return 1}
        if(a<135){return 2}
        if(a<180){return 3}
        if(a<225){return 4}
    }
    if(b==46){
        if(a<92){return 1}
        if(a<138){return 2}
        if(a<184){return 3}
        if(a<230){return 4}
    }
    if(b==47){
        if(a<94){return 1}
        if(a<141){return 2}
        if(a<188){return 3}
        if(a<235){return 4}
    }
    if(b==48){
        if(a<96){return 1}
        if(a<144){return 2}
        if(a<192){return 3}
        if(a<240){return 4}
    }
    if(b==49){
        if(a<98){return 1}
        if(a<147){return 2}
        if(a<196){return 3}
        if(a<245){return 4}
    }
    if(b==50){
        if(a<100){return 1}
        if(a<150){return 2}
        if(a<200){return 3}
        if(a<250){return 4}
    }
    if(b==51){
        if(a<102){return 1}
        if(a<153){return 2}
        if(a<204){return 3}
    }
    if(b==52){
        if(a<104){return 1}
        if(a<156){return 2}
        if(a<208){return 3}
    }
    if(b==53){
        if(a<106){return 1}
        if(a<159){return 2}
        if(a<212){return 3}
    }
    if(b==54){
        if(a<108){return 1}
        if(a<162){return 2}
        if(a<216){return 3}
    }
    if(b==55){
        if(a<110){return 1}
        if(a<165){return 2}
        if(a<220){return 3}
    }
    if(b==56){
        if(a<112){return 1}
        if(a<168){return 2}
        if(a<224){return 3}
    }
    if(b==57){
        if(a<114){return 1}
        if(a<171){return 2}
        if(a<228){return 3}
    }
    if(b==58){
        if(a<116){return 1}
        if(a<174){return 2}
        if(a<232){return 3}
    }
    if(b==59){
        if(a<118){return 1}
        if(a<177){return 2}
        if(a<236){return 3}
    }
    if(b==60){
        if(a<120){return 1}
        if(a<180){return 2}
        if(a<240){return 3}
    }
    if(b==61){
        if(a<122){return 1}
        if(a<183){return 2}
        if(a<244){return 3}
    }
    if(b==62){
        if(a<124){return 1}
        if(a<186){return 2}
        if(a<248){return 3}
    }
    if(b==63){
        if(a<126){return 1}
        if(a<189){return 2}
        if(a<252){return 3}
    }
    if(b==64){
        if(a<128){return 1}
        if(a<192){return 2}
        if(a<256){return 3}
    }
    if(b==65){
        if(a<130){return 1}
        if(a<195){return 2}
        if(a<260){return 3}
    }
    if(b==66){
        if(a<132){return 1}
        if(a<198){return 2}
        if(a<264){return 3}
    }
    if(b==67){
        if(a<134){return 1}
        if(a<201){return 2}
    }
    if(b==68){
        if(a<136){return 1}
        if(a<204){return 2}
    }
    if(b==69){
        if(a<138){return 1}
        if(a<207){return 2}
    }
    if(b==70){
        if(a<140){return 1}
        if(a<210){return 2}
    }
    if(b==71){
        if(a<142){return 1}
        if(a<213){return 2}
    }
    if(b==72){
        if(a<144){return 1}
        if(a<216){return 2}
    }
    if(b==73){
        if(a<146){return 1}
        if(a<219){return 2}
    }
    if(b==74){
        if(a<148){return 1}
        if(a<222){return 2}
    }
    if(b==75){
        if(a<150){return 1}
        if(a<225){return 2}
    }
    if(b==76){
        if(a<152){return 1}
        if(a<228){return 2}
    }
    if(b==77){
        if(a<154){return 1}
        if(a<231){return 2}
    }
    if(b==78){
        if(a<156){return 1}
        if(a<234){return 2}
    }
    if(b==79){
        if(a<158){return 1}
        if(a<237){return 2}
    }
    if(b==80){
        if(a<160){return 1}
        if(a<240){return 2}
    }
    if(b==81){
        if(a<162){return 1}
        if(a<243){return 2}
    }
    if(b==82){
        if(a<164){return 1}
        if(a<246){return 2}
    }
    if(b==83){
        if(a<166){return 1}
        if(a<249){return 2}
    }
    if(b==84){
        if(a<168){return 1}
        if(a<252){return 2}
    }
    if(b==85){
        if(a<170){return 1}
        if(a<255){return 2}
    }
    if(b==86){
        if(a<172){return 1}
        if(a<258){return 2}
    }
    if(b==87){
        if(a<174){return 1}
        if(a<261){return 2}
    }
    if(b==88){
        if(a<176){return 1}
        if(a<264){return 2}
    }
    if(b==89){
        if(a<178){return 1}
        if(a<267){return 2}
    }
    if(b==90){
        if(a<180){return 1}
        if(a<270){return 2}
    }
    if(b==91){
        if(a<182){return 1}
        if(a<273){return 2}
    }
    if(b==92){
        if(a<184){return 1}
        if(a<276){return 2}
    }
    if(b==93){
        if(a<186){return 1}
        if(a<279){return 2}
    }
    if(b==94){
        if(a<188){return 1}
        if(a<282){return 2}
    }
    if(b==95){
        if(a<190){return 1}
        if(a<285){return 2}
    }
    if(b==96){
        if(a<192){return 1}
        if(a<288){return 2}
    }
    if(b==97){
        if(a<194){return 1}
        if(a<291){return 2}
    }
    if(b==98){
        if(a<196){return 1}
        if(a<294){return 2}
    }
    if(b==99){
        if(a<198){return 1}
        if(a<297){return 2}
    }
    if(b==100){
        if(a<200){return 1}
        if(a<300){return 2}
    }
    if(b<=200 && a<=200){
        return 1
    }
    var result=0
    var counter=b
    for(;a>=counter;counter=add(counter,b)){
        result=add(result,1)
    }
    return result
}

Tcl, 0 additions.

Well, I had to find a way that does not use other data structures but is still not what you want:

# coroutine counter.
proc ccnt {} {yield [info level]; ccnt}
# add implementation without add.
proc cadd {a b} {
    set last 2
    coroutine cadda ccnt
    coroutine caddb ccnt
    while {[cadda]<=$a} {}
    while {[caddb]<=$b} {set last [cadda]}
    rename cadda {}
    rename caddb {}
    return $last
}

proc divide {a b {c 0}} {
    if {$c == 0} {set c $b} {set c [cadd $b $c]}
    if {$c>$a} {tailcall info level}
    divide $a $b $c
}

Uses the current stack size of different green threads.

Python, 320703 additions

def divide(a, b):
    quotient = 0
    c = 0
    d = 0
    while add(d, b) <= a:
        c = add(c, 1)
        d = add(d, b)
    return c

As always, a last-place reference answer. This simply adds 1 to a "quotient" and b to a "remultiplication" variable until it hits a.

Here is the debugging code:

add_used = 0

def add(a, b):
    global add_used
    add_used += 1
    return a + b

for a in range(1, 201):
    for b in range(1, a+1):
        print "%d / %d = %d" % (a, b, divide(a, b))

print "Additions used:", add_used

Tcl, 0 additions

proc divide {a b} {
    set sa [string repeat . $a]
    set sb [string repeat . $b]
    set sr ""
    while 1 {
        append sc $sb
        if {[string le $sc]>[string le $sa]} break
        append sr .
    }
    return [string le $sr]
}

Why not use strings?