dc, 37

?[16~rd0<m]dsmxk[A~r17*+48+Pz0<p]dspx

Recursively divmods by 16, pushing the remainder to the stack until nothing left to divide. Then print each element of the stack, using divmod by 10 to achieve A-F digits. Probably more detail tomorrow... (and hopefully less bytes).

Try it online!

Bash (function), 62

Thanks to @manatwork for suggesting using recursion.

h()(x=({0..9} {A..F})
echo `(($1>15))&&h $[$1/16]`${x[$1%16]})

Try it online!

sed, 341 bytes

:
s/\b/_/2
s/[13579]/&;/g
y/123456789/011223344/
s/;0/5/g
s/;1/6/g
s/;2/7/g
s/;3/8/g
s/;4/9/g
s/;_;_;_;_/=/
s/;_;_;__/+/
s/;_;__;_/:/
s/;_;___/>/
s/;__;_;_/</
s/;__;__/?/
s/;___;_/(/
s/;____/*/
s/_;_;_;_/-/
s/_;_;__/^/
s/_;__;_/%/
s/_;___/$/
s/__;_;_/#/
s/__;__/@/
s/___;_/!/
s/____/)/
/[1-9_]/b
y/)!@#$%^-*(?<>:+=/0123456789ABCDEF/
s/^0*//

It's not the obvious language for this challenge, but it does have the advantage of supporting input numbers up to (depending on your implementation) between 4000 digits and the limit of your system's available (virtual) memory. I converted RSA-1024 to hex in about 0.6 seconds, so it scales reasonably well.

It works using successive division by two, accumulating every 4 bits of carry into a hex digit. We use non-letter characters to represent our output, so that we always accumulate carry between the decimal input and the hex output, and convert to conventional hexadecimal at the very end.

Uiua, 24 bytes

⇌⊏:⊂+@0⇡10+@A⇡6°⋯⬚0↯∞_4⋯

Try it!

⇌⊏:⊂+@0⇡10+@A⇡6°⋯⬚0↯∞_4⋯
                       ⋯ # convert to binary 
                 ⬚0↯∞_4  # split into nybbles
               °⋯        # convert back to numbers
 ⊏:                      # index from
   ⊂+@0⇡10+@A⇡6          # 0123456789ABCDEF
⇌                        # reverse

-3 bytes using builtins (Europe2048):

⇌⊏:HexDigits°⋯⬚0↯∞_4⋯

Japt, 11 bytes

;s16ogEf"%w

Try it

;s16ogEfwi%     :Implicit input of integer
 s              :Convert to string in custom base
  16o           :  Range [0,16)
     g          :  Index (0-based) each into
;     E         :    ASCII
       f        :    Match
        "%w     :      Japt RegEx class \w, which is /[a-z0-9]/gi

Retina 0.8.2, 56 bytes

.+
$*#;
+`(#+)\1{15}
$1;
(#{10})?(#*);
$1$.2
T`#d`_L`#+.

Try it online! Link includes test cases. Retina doesn't actually have a built-in for this anyway. Explanation:

.+
$*#;

Convert the input to unary using #s and append a ;.

+`(#+)\1{15}
$1;

Keep trying to divide the input by 16, leaving the remainder after another ;.

(#{10})?(#*);
$1$.2

Convert each remainder to base 10, but if it's 10 or more, then just convert the last digit, leaving the 10 #s behind.

T`#d`_L`#+.

Change the digits after #s from 0-5 to A-F and delete the #s.

Charcoal, 25 bytes

Ｎθ≔⁺⭆χι…α⁶ηＰ0Ｗθ«←§ηθ≧÷¹⁶θ

Try it online! Link is to verbose version of code. Explanation:

Ｎθ

Input the number.

≔⁺⭆χι…α⁶η

Create the string of hexadecimal digits by joining the range 0..9 with the first six letters.

Ｐ0

Output a zero in case the input is zero.

Ｗθ«

Repeat while the number is nonzero.

←§ηθ

Cyclically index into the hex digits and move the cursor to the left after printing.

≧÷¹⁶θ

Integer divide the number by 16.

APL(NARS), chars 34, bytes 68

{⍵≤0:,'0'⋄(∇⌊⍵÷16),(1+16∣⍵)⊃⎕D,⎕A}

test:

  g←{⍵≤0:,'0'⋄(∇⌊⍵÷16),(1+16∣⍵)⊃⎕D,⎕A}
  g 0
0
  g 100
064
  g 1000
03E8
  g 1
01

PHP, 65 66 64+1 62 59 bytes

function h($n){$n&&h($n>>4);echo"0123456789abcdef"[$n&15];}

recursive printing function, prints a leading zero (insert >16 before && to remove it)

programs, 64 bytes +1 for -R (run as pipe with -nR)

for(;$n=&$argn;$n>>=4)$s="0123456789abcdef"[$n&15].$s;echo$s?:0;

requires PHP 5.6 or later (5.5 cannot index string literals)

or

for(;$n=&$argn;$n>>=4)$s=(abcdef[$n%16-10]?:$n%16).$s;echo$s?:0;

requires PHP 5.6 or 7.0 (7.1 understands negative string indexes)

Run as pipe with -nR or try them online.

C (gcc), 45 44 bytes

f(n){n&&f(n/16);n%=16;putchar(n+48+n/10*7);}

Try it online!

><>, 46 + 3 = 49 bytes

This would have been shorter if ><> had integer division, which we now have to emulate by subtracting modulo 1. Still, I think this uses some pretty neat wrapping around tricks!

>:?!v:f1+%:a(?v  v
\-%1:,+1f}+"0"<+7<
!?:r/ro;

Try it online!

Explanation

First loop

>:?!v:f1+%:a(?v  v
\-%1:,+1f}+"0"<+7<

The first loop performs the classic converting to hex algorithm. It does modulo 16 (:f1+%) and checks if the result is < 10 (:a(?). If it's not, we need to add 7 (7+) in order to go from the decimals to the capital alphabet in the ASCII table. Else, we can proceed by adding the ASCII value for 0 ("0"+) and shifting the character to be output to the bottom of the stack because we'll have to output them in reverse order. The top value is then replaced by its result of integer division by 16. This is emulated by computing a/b - (a/b)%1 (f1+,:1%-). When the loop is finished, the stack contains the hexadecimal characters in reversed output order and a 0.

Second loop

!?:r<ro;

The second loop reverses the list and checks if top element is 0. If it is, we know all nonzero were printed and we should terminate. Else, we output the character and reverse the list again to prepare for the next iteration. The : when entering the second loop will duplicate the 0 which has no effect.

C, 48 bytes

h(x){x/16&&h(x/16);x%=16;putchar(x+=48+x/10*7);}

This is not completely original, I shaved 5 bytes off of the version Digital Trauma put up.

REXX, 80 78 bytes

arg n
h=
do while n>0
  h=substr('0123456789ABCDEF',n//16+1,1)h
  n=n%16
  end
say h

Ruby, 40 bytes

Stolen from Inspired by manatwork's answer, but using an interesting loophole to make it shorter.

h=->n{(n>15?h[n/16]:'')+(n%16).to_s(17)}

C (clang), 83 bytes

f(n){char h[8]={0};int i=8;while(i>0){h[i-1]=n%16;n/=16;printf("%x",h[8-i]);i--;}}

Try it online!

Alternate solution in C

APL (Dyalog APL), 17 bytes

Must be run with ⎕IO←0, which is default on many APL systems.

(⎕D,⎕A)[16⊥⍣¯1⊢⎕]

Try it online!

(⎕D,⎕A)[…] Digits concatenated to Alphabet, then indexed by…

 16⊥⍣¯1  the inverse of 16-Base-to-Number, i.e. Number-to-Base-16

 ⊢ applied to

 ⎕ numeric input

Befunge-93, 58

&:88+%"0"+:"9"`7*+\8/2/:!#|_#
,_@                       >:#

First time doing a real golfing challenge in Befunge, I bet there's a one-liner for this that's shorter since all those spaces in the middle of the second line seem wasteful.

You can step through it here. Partial explanation:

&: Take input.

:88+%: Take the remainder modulo 16.

"0"+: Add it to the ASCII value of 0.

:"9"`: If the result is greater than the ASCII value of 9...

7*+: Add 7 to convert it to a letter.

\: Save the resulting character on the stack.

8/2/: Divide by 16 rounding down.

:!#|_: Exit the loop if the result is 0.

#: Otherwise go back to the modulus step.

>:#,_@ (wrapping around): Once finished, output the stack in LIFO order.

Perl 6,  53  48 bytes

{[R~] (0..9,'A'..'F').flat[($_,*div 16...^0)X%16]||0}
{[R~] (0..9,'A'..'F').flat[.polymod(16 xx*)]||0}

This creates a sequence of values that are Integer divided (div), until the result is 0 excluding the 0 from the sequence

$_, * div 16 ...^ 0

It then crosses (X) that sequence using the modulus operator (%) with 16

( … ) X[%] 16

It uses those values as indexes into a flattened list consisting of two Ranges 0..9 and 'A'..'Z'

( 0 .. 9, 'A' .. 'Z' ).flat[ … ]

Finally it concatenates (~) them using the reverse (R) meta operator

[R[~]] …

If that results in a False value (empty string), return 0

… || 0

Usage:

# (optional) give it a lexical name for ease of use
my &code = { … }

say <15 1000 256 0>.map: &code;
# (F 3E8 100 0)

say code 10¹⁰⁰;
# 1249AD2594C37CEB0B2784C4CE0BF38ACE408E211A7CAAB24308A82E8F10000000000000000000000000

Javascript ES6, 64 58 bytes

v=>eval('for(z="";v;v>>=4)z="0123456789ABCDEF"[v%16]+z')

Saved 6 bytes thanks to ןnɟuɐɯɹɐןoɯ and user81655.

Haskell, 59 58 43 41 39 bytes

s="0123456789ABCDEF"
(sequence(s<$s)!!)

Usage example: sequence(s<$s)!!) $ 1000 -> "00000000000003E8".

This creates a list of all hexadecimal numbers up to 16 hex-digits. Luckily this happens in order, so we can simply pick the nth one.

Edit: @Mauris squeezed out 2 bytes. Thanks!

Python, 59 58 bytes

h=lambda i:(i>15 and h(i/16)or'')+"0123456789abcdef"[i%16]

1 byte saved by CarpetPython

Run as: print h(15)

Test it here (Ideone.com).

Explanation:

h=lambda i:                                                 # Define h as a function that takes two arguments
           (i>15 and h(i/16)or'')                           # Evaluate h(i/16) if i > 15, else, give ''
                                 +"0123456789abcdef"[i%16]  # Append (i%16)'th hexadecimal number.

𝔼𝕊𝕄𝕚𝕟, 31 chars / 62 bytes

↺a=⬯;ï;ï≫4@a=⩥ḊĀⒸª⩥⁽ṁṇ⸩⨝[ï%Ḑ]+a

Try it here (Firefox only).

Okay, I figured out some more stuff that golfed it down.

Explanation

It's essentially the same solution as @SuperJedi224's ES6 solution - but with something different.

See ⩥ḊĀⒸª⩥⁽ṁṇ⸩⨝? That's a really fancy way of writing "0123456789ABCDEF". ⩥Ḋ creates a range from 0 to 10, Ⓒª⩥⁽ṁṇ⸩ creates a range from 65 to 71 and converts it to a string of ASCII, and Ā...⨝ concatenates the two ranges and joins them into one string. This was probably the coolest part of my solution.

Bonus non-competitive version, 24 chars / 45 bytes

↺;ï;ï≫4@ᵴ=(⩥Ḋ⨝+ᶐ)[ï%Ḑ]+ᵴ

I decided to add an alphabet string, like in Pyth.

Seriously, 35 bytes

,`;4ª@%)4ª@\`╬Xε D`@;7ªD+@9<7*+c+`n

Hex Dump:

2c603b34a640252934a6405c60ce58ee204460403b37a6442b40393c372a2b632b606e

Try It Online

Explanation:

,                                    Get evaluated input
 `          `╬                       Repeat the quoted function until top of stack is 0
  ;4ª@%                              Make a copy of the number mod 16
       )                             Send it to bottom of stack
        4ª@\                         Integer divide the original copy by 16
              X                      Delete the leftover zero. At this point the stack is 
                                     the "digits" of the hex number from LSD to MSD
               ε                     Push empty string
                 D`              `n  Essentially fold the quoted function over the stack.
                   @;                Roll up the next lowest digit, make a copy
                     7ªD+            Add 48
                         @           Bring up the other copy
                          9<         1 if it's at least 10, else 0
                            7*       Multiply with 7. 
                              +      Add. This will shift 58->65 and so on.
                               c     Convert to character.
                                +    Prepend to current string.

Note that the ;7ªD+@9<7*+c is equivalent to 4ª▀E, which would save 8 bytes, but I thought that perhaps a function that pushes the base b digits as a string might be considered too much of a "heaxadecimal built-in".

Pyth, 33 26 21 20 bytes

This was a fun one.

sm@+jkUTGi_d2_c_.BQ4

Try it online.

Explained:

                .BQ      Convert input to a binary string, e.g. 26 -> '11010'
             _c_   4     Reverse, chop into chunks of 4, and reverse again. We reverse 
                         because chop gives a shorter last element, and we want a shorter
                         first element: ['1', '0101']
                         Reversing three times is still shorter than using .[d4 to pad the
                         binary string to a multiple of 4 with spaces.
 m                       Map across this list:
         i_d2                Take the value of the reversed string in binary,
  @                          and use it as an index into the string:
   +jkUTG                    '0123456789abcdefghijklmnopqrstuvwxyz'
                             (The alphabet appended to the range 0 to 10)
s                        Concatenate to create the final string.

CJam, 22 21 bytes

ri{Gmd_A<70s=+\}h;]W%

Thanks to @MartinBüttner for golfing off 1 byte!

Try it online!

How it works

ri                      e# Read an integer from STDIN.
  {             }h      e# Do:
   Gmd                  e#   Push qotient and residue of the division by 16.
      _A<               e#   Check if the residue is less than 10.
         70s            e#   Push "70".
            =           e#   Select the character that corresponds to the Boolean.
             +          e#   Add the character to the digit.
                        e#   This way, 10 -> 'A', etc.
               \        e#   Swap the quotient on top of the stack.
                        e# While the quotient is non-zero, repeat the loop.
                  ;     e# Pop the last quotient.
                   ]W%  e# Reverse the stack.

Javascript, 49 43 bytes.

h=i=>(i?h(i>>4):0)+"0123456789abcdef"[i%16]

6 bytes saved by user81655.

Test it here.

This has two leading zeroes, which is allowed by the rules.

Here's a version without leading zeroes: (47 bytes).

h=i=>(i>15?h(i>>4):"")+"0123456789abcdef"[i%16]

Test it here.

Both of these uses exactly the same approach as my Python answer.

Turing Machine Code, 412 bytes

As usual, I'm using the rule table syntax defined here. You can test it on that site or, alternatively, using this java implementation.

0 * * l B
B * * l C
C * 0 r D
D * * r E
E * * r A
A _ * l 1
A * * r *
1 0 9 l 1
1 1 0 l 2
1 2 1 l 2
1 3 2 l 2
1 4 3 l 2
1 5 4 l 2
1 6 5 l 2
1 7 6 l 2
1 8 7 l 2
1 9 8 l 2
1 _ * r Y
Y * * * X
X * _ r X
X _ _ * halt
2 * * l 2
2 _ _ l 3
3 * 1 r 4
3 1 2 r 4
3 2 3 r 4
3 3 4 r 4
3 4 5 r 4
3 5 6 r 4
3 6 7 r 4
3 7 8 r 4
3 8 9 r 4
3 9 A r 4
3 A B r 4
3 B C r 4
3 C D r 4
3 D E r 4
3 E F r 4
3 F 0 l 3
4 * * r 4
4 _ _ r A

Counts down from the input in base 10 while counting up from 0 in base 16. On decrementing zero, it erases the input block and terminates.

SpecBAS - 110 bytes

1 h$="0123456789ABCDEF",r$=""
2 INPUT d
4 q=INT(d/16),r=d-(q*16),r$=h$(r+1)+r$,d=q
5 IF q>15 THEN 4
6  ?h$(q+1)+r$

This uses an algorithm I found on WikiHow (2nd method).

Strings in SpecBAS are 1-based, hence the +1 to pick out the correct element.

Ruby, 48 characters

(Copy of Loovjo's Python answer.)

h=->n{(n>15?h[n/16]:'')+[*?0..?9,*?a..?f][n%16]}

Sample run:

2.1.5 :001 > h=->n{(n>15?h[n/16]:'')+[*?0..?9,*?a..?f][n%16]}
 => #<Proc:0x00000001404a38@(irb):1 (lambda)> 
2.1.5 :002 > h[15]
 => "f" 
2.1.5 :003 > h[1000]
 => "3e8" 
2.1.5 :004 > h[256]
 => "100" 

Pyre, 98 bytes

Doing this in a language without arithmetic operators was probably a mistake.

let h=def (n)(if n.gt(15)h(n.div(16).int!)else "").concat("0123456789abcdef".list!.get(n.mod(16)))

Use like this:

do
  let h = ...
  print(h(15))
end

Ungolfed:

let h = def (n) do
    if n.gt(15) 
        let x = h(n.div(16).int!)
    else 
        let x = ""
    x.concat("0123456789abcdef".list!.get(n.mod(16)))
end

C (function), 51

Recursive function takes input integer as a parameter:

f(n){n>>4?f(n>>4):0;n&=15;n+=n>9?55:48;putchar(n);}

Test driver:

#include <stdio.h>

f(n){if(n>>4)f(n>>4);n&=15;n+=n<10?48:55;putchar(n);}

int main (int argc, char **argv) {

    f(15);puts("");
    f(1000);puts("");
    f(256);puts("");
    f(0);puts("");

    return 0;
}

MATL, 27 bytes

i`16H#\wt9>?7+]wt]xN$hP48+c

This uses release 5.1.0 of the language/compiler, which is earlier than this challenge.

Example

>> matl
 > i`16H#\wt9>?7+]wt]xN$hP48+c
 >
> 1000
3E8

Explanation

i              % input number
`              % do...
  16H#\        % remainder and quotient of division by 16
  w            % move remainder to top of stack
  t9>          % does it exceed 9?
  ?            % if so
    7+         % add 7 (for letter ASCII code)
  ]            % end if
  w            % move quotient back to the top
  t            % duplicate 
]              % ...while (duplicated) quotient is not zero
x              % delete last quotient (zero)
N$h            % create vector of all remainders 
P              % flip vector
48+c           % add 48 and convert to char (will be implicitly displayed)

Julia, 55 bytes

h(n)=(n>15?h(n÷16):"")"0123456789ABCDEF"[(i=n%16+1):i]

This is the basic recursive function implementation. It accepts an integer and returns a string.

If the input is less than 15, floor divide it by 16 and recurse, otherwise take the empty string. Tack this onto the front of the appropriately selected hexadecimal character.

Java, 92 89 bytes

String x(int v){String z="";for(;v>0;v/=16)z="0123456789ABCDEF".charAt(v%16)+z;return z;}