Gol><>, 35 bytes

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

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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

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