Tcl, 113 bytes

proc S n {set r [expr $n*($n+1)/2]
while \$r>9 {set r [join [split [expr [join [split $r ""] *]] ""] +]}
expr $r}

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J-uby, 38 bytes

This is me resisting forking J-uby to add a bunch of single-character aliases for useful functions.

:+|:sum|:!~&((d=:digits)|:/&:*|d|:sum)

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Explanation

:+ | :sum | :!~ & ((d = :digits) | :/ & :* | d | :sum)

:+ | :sum |                                             # Range 1..n, then sum, then
            :!~ & (                                  )  # Until a fixed point is reached...
                  ((d = :digits) |                      #   Get digits (while saving the function to d for later), then
                                   :/ & :* |            #   Reduce with product, then
                                             d | :sum   #   Get digits, then sum

APL (Dyalog Unicode), 16 bytes

-4 thanks to @user

f←{⍎¨⍕×/⍎¨⍕+/⍵}⍣≡⍳

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Julia 0.6, 56 bytes

f(n,k=(n+1)n÷2)=k>9?f(0,sum(digits(prod(digits(k))))):k

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Pretty straightforward: calculate (n+1)n÷2 for sum from 1..n, check if it's a single digit number (>9), if it's not, try again with k set to the sum of the digits of the product of the digits of k, else return k.

Japt, 16 14 13 bytes

_ì ×ìx}gN®õ x

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Explanation

                  :Implicit input of integer U
         ®        :Map
        N         :The array of inputs (which just contains U)
          õ       :  Range [1,U]
            x     :  Reduce by addition
_     }g          :Take the last element of N, run it through the following function and push the result to N
                  : Repeat U times and then return the last element of N
 ì                :  Split to an array of digits
   ×              :  Reduce by multiplication
    ìx            :  Split to an array of digits and reduce by addition

Gol><>, ₃₅ 33 bytes

1AY:P*2,TYMR*YR+:a(?Bt
:a(qlBaSD$

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-2 bytes by Jo King.

Extensive use of functions and implicit infinite loops.

Example full program & How it works

1AGIE;GN
2AY:P*2,YlMR*YlR+:a(?B8R!
:a(?BaSD$

<main program>
1AG       Register row 1 as function G
   IE;    Take number input; halt on EOF
      GN  Call G and print the result as number
          Repeat indefinitely

<function G>
2AY            Register row 2 as function Y
   :P*2,       Sum of 1 to n
        Y      Call Y (break into digits)
         lMR*  Product
Y              Call Y
 lR+           Sum (an implicit zero is used)
    :a(?B      Return if the result is less than 10
         8R!   Skip initial 8 commands
               Repeat indefinitely

<function Y>
:a(?B      Return if the top is less than 10
     aSD   Divmod by 10; [... n] => [... n/10 n%10]
        $  Swap top two, so the "div" goes to the top

Befunge, 136 bytes

101p&::*+2/>:82+%01g*01p82+/:#v_$01gv
X      v_v# #:/+82p10+g10%+82: <p100<
v:g10$ >#<#^                 #<^
>82+/#v_.@
      >101p^

You can try it here.

While not all interpreters have a large enough cell size, it works with small numbers for pretty much anyone out there. For a larger number of n you might need a interpreter like BefunExec.

Java 8, 129 bytes

n->{long r=1,i=n;for(;i>1;r+=i--);for(;r>9;r=(i+"").chars().map(p->p-48).sum(),i=1)for(int s:(r+"").getBytes())i*=s-48;return r;}

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Explanation:

n->{            // Method with integer parameter and long return-type
  long r=1,     //  Result-long, starting at 1
       i=n;     //  Temp integer, starting at the input `n`
  for(;i>1;     //  Loop as long as `i` is not 1 yet
      r+=i--);  //   And increase `r` by `i`
  for(;r>9      //  Loop as long as `r` is not a single digit yet
      ;         //    After every iteration:
       r=(i+"").chars().map(p->p-48).sum(),
                //     Set `r` to the sum of digits of `i`
       i=1)     //     And reset `i` to 1
    for(int s:(r+"").getBytes())i*=s-48;
                //    Set `i` to the product of the digits of `r`
  return r;}    //  Return `r` as result

Ruby, 57 bytes

->n{("*+"*n).chars.reduce(-~n*n/2){|x,y|eval x.digits*y}}

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R, 152 130 109 bytes

function(w,x=w*(w+1)/2,y=prod(d(x)),z=sum(d(y)))"if"(z>9,f(,z),z)
d=function(x)x%/%10^(0:max(0,log10(x)))%%10

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@Giuseppe found 21 42 bytes with various R things I'm not used to yet, along with a way to get the digits of a number without coercing to string and back, and with fewer bytes!

# Old
d=function(x)strtoi(el(strsplit(paste(x),"")))
# New
d=function(x)x%/%10^(0:max(0,log10(x)))%%10

options(scipen=9) is was required for the case of 9854 for the old function, because the first product stage ends up as 80000, which R prints as 8e+05.

Python 2, 77 72 71 62 60 bytes

lambda n:reduce(lambda x,c:eval(c.join(`x`)),'*+'*n,-n*~n/2)

Thanks to @xnor for golfing off 2 bytes!

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F#, 175 bytes

let d n=seq{for i in(string n).ToCharArray() do yield string i|>uint64}
let c n=
 let mutable r=Seq.sum{1UL..n}
 while r>9UL do r<-d r|>Seq.reduce(fun a x->x*a)|>d|>Seq.sum
 r

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The only caveat to the function is that the input value must be of type uint64.

Ungolfed it's a little like this:

let d n=seq{for i in(string n).ToCharArray() do yield string i|>uint64}

let c n =
 let mutable r = Seq.sum {1UL..n}
 while r > 9UL do
  r<-d r
  |> Seq.reduce(fun a x->x*a)
  |> d
  |> Seq.sum
 r

The function d n converts the number n into its component digits. It first converts to a string, then gets each character in the string. Each character must then be converted back into a string, otherwise the characters will be converted to their ASCII values instead of their "real" values.

The c n function is the main function, with n as the initial value. In this function r is our running value. The while loop does the following:

• Convert r into its component digits (d r).
• Get the product of all those digits. This uses Seq.reduce which takes a function with the accumulated value (a) and the next value in the sequence (x) and in this case returns the product. The initial value is the first element in the sequence.
• Convert this product value into its component digits (d).
• Sum the digits from before, and assign this to r.

Gaia, 8 bytes

┅⟨Σ₸∨Π⟩°

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The old explanation (before fixing a bug that is Gaia’s fault IMO :P):

┅⟨ΣΠ⟩° – Full program. N = The input.
┅      – Range. Push [1, N] ⋂ ℤ to the stack.
 ⟨  ⟩° – While no two consecutive iterations yield the same result, do:
  Σ    – Sum (or digital sum, when applied to an integer).
   Π   – Digital product.

Saved 1 byte thanks to Dennis.

Charcoal, 18 bytes

≔Σ…·¹ＮθＷ›θ⁹≔ΣΠθθＩθ

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

≔Σ…·¹Ｎθ

Sum the integers up to the input.

 Ｗ›θ⁹≔ΣΠθθ

While the result is greater than 9, take the sum of digits of the product of digits.

Ｉθ

Cast the result to string and implicitly print it.

Perl 5 -p, 61 bytes

$_*=$_/2+.5;$_=eval((eval s/./$&*/gr.1)=~s/./+$&/gr)while$_>9

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PowerShell Core, 91 101 93 bytes

Function F($a){$o=$a*($a+1)/2;1,2|%{$o=[char[]]"$([char[]]"$o"-join'*'|iex)"-join'+'|iex};$o}

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Ungolfed a little...

Function F ($a)
{
    $o=$a*($a+1)/2;
    1..2 | % {
        $o = [char[]]"$o"-join '*' | iex;
        $o = [char[]]"$o"-join '+' | iex;
    }
    $o | Write-Output
}

First steps were to split the integers into digits -- did this by splitting the integer into an array of stringscharacters. Afterwards, insert the operand, and then evaluate the string as a command. Then, it's a matter of doing the multiple-add cycle until the input is one digit.

iex is an alias for Invoke-Command which evaluates a string passed into the first param position.

Edit: As requested by @AdmBorkBork, I have added a function header to the byte count. Also, I did a little math and realized that an upper bound on the number of iterations is < log log 10^6 < log 6 < 2, so that saved another six bytes.

Edit x2: @AdmBorkBork found a more terse way of converting the integer into a math expression, and then suggested piping it into iex. This saved 8 bytes. Thank you!

Perl 6, 49 bytes

{($_*.succ/2,{[+] ([*] .comb).comb}...9>=*).tail}

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Haskell, 72 71 63 bytes

g=map(read.pure).show
f n=until(<10)(sum.g.product.g)$sum[1..n]

Thanks to @BMO for a byte and @nimi for 8 bytes!

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Stax, 14 13 10 bytes

ñu┌↕a√äJ²┐

Run and debug it

Was pretty fun to make. I wonder if there is a more concise way to do the comparison at the end.

Explanation

|+wE:*E|+c9>                 # Full Program Unpacked
|+                           # Create range and sum it
   wE:*                      # Start loop, digitize number, product of digits
       E|+                   # Digitize number, sum digits
          c9>                # Duplicate, check length is = 1
                             # Otherwise loop back to the 'w' character

-1 bytes thanks to ovs

-3 bytes thanks to Scrooble

05AB1E, 7 bytes

LOΔSPSO

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Exlpanation

L         # push range [1 ... input]
 O        # sum range
  Δ       # loop until top of stack stops changing
   SP     # product of digits
     SO   # sum of digits

PHP 7, 89 bytes

for($a=$argn*-~+$argn/2;$a>9;)$a=array_sum(($s=str_split)(array_product($s($a))));echo$a;

Run as pipe with -r or try it online.

• PHP always takes input as string, so I have to use + to cast to int for ~ to work as wanted.
• Pre-increment would not work: no matter where I put it, it would effect both operands.
• But: It doesn´t matter if the single digit takes place before or after the iteration (additional iterations wouldn´t change a thing); so I can use for() instead of do ... while().
• PHP 7 or later is required for the inline assignment of the function name.
  Older PHP requires one more byte: for($s=str_split,$a=...;$a>9;)$a=array_sum($s(...));
  (Not assigning str_split to a variable at all would waste another byte.)

JavaScript (ES6), 60 bytes

f=(n,k=n*++n/2)=>k>9?f(!n,eval([...k+''].join('*+'[+!n]))):k

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Commented

f = (                     // f = recursive function taking:
  n,                      //   n = original input
  k = n * ++n / 2         //   k = current value, initialized to sum(i=1..n)(i)
) =>                      //
  k > 9 ?                 // if k has more than 1 digit:
    f(                    //   recursive call to f() with:
      !n,                 //     a logical NOT applied to n
      eval(               //     the result of the expression built by:
        [...k + '']       //       turning k into a list of digits
        .join('*+'[+!n])  //       joining with '*' on even iterations or '+' on odd ones
      )                   //     end of eval()
    )                     //   end of recursive call
  :                       // else:
    k                     //   stop recursion and return the last value

Alternate version, 59 bytes (non-competing)

A non-recursive version that only works for n < 236172. (It covers the requested range but does not qualify as a valid generic algorithm.)

n=>[...'*+*+'].map(o=>n=eval([...n+''].join(o)),n*=++n/2)|n

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Pyth, 11 bytes

usj*FjGTTsS

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usj*FjGTTsS – Full program. N = The input.
          S – Range. Yield [1, N] ⋂ ℤ.
         s  – Sum.
u           – While no two consecutive iterations yield the same result, do (var: G):
   *FjGT    – Digital product.
 sj     T   – Digital sum.

MATL, 15 13 bytes

In tribute to the Language of the month:

:`sV!UpV!Utnq

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I don't think there's a simpler way to get the digits of a number than to convert the number to a string V, then transposing it !, and converting this vertical vector back to a numeric one U.

Saved 2 bytes thanks to the Creator¹ himself! I forgot the implicit end, meaning I could remove ], and instead of comparing the number of elements with 1, I could simply decrement that value and use it as a boolean directly.

So, the explanation goes like this:

                 % Grab input n implicitly
:                % Range from 1 ... n inclusive
 `               % Do ... while
  s               % sum the vector
   V!U            % Convert the number to digits
      p           % Take the product of these digits
       V!U        % Convert the product into digits
          t       % Duplicate the result
           n      % Count the number of elements
            q     % Decrement the number of elements
                  % Loop until the number of elements is 1
                 % Implicit end

¹... of MATL, Luis Mendo.

Husk, 7 bytes

ωöΣdΠdΣ

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Jelly, 8 bytes

RSDPDƲÐL

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Full program (it returns a singleton array containing the result, but the brackets aren't visible in STDOUT).