# [C# (Visual C# Interactive Compiler)], 113 bytes

<!-- language-all: lang-cs -->

    s=>{var t=s;do{s=t;t="";for(int i=0;i<s.Length;)t+=(s[i]-48)*(s[i++]!=(s+0)[i]?1:2*++i/i);}while(t!=s);return t;}

[Try it online!][TIO-jqgnrb8r]

[C# (Visual C# Interactive Compiler)]: http://www.mono-project.com/docs/about-mono/releases/5.0.0/#csc
[TIO-jqgnrb8r]: https://tio.run/##RU47b8IwEN7zK46og40NjU1ADsZ068TeATGE1IClylTxUSpF@e2pQ4Dect/rHlWYVMF17xdfrQLWzh/50NZwANMFs25@yhrQBP15boJBjSZN9eFcE@cRnMm0W4XpxvojnjRFZkjYut0kV3TcI8Z2oyixjEb1TSzlmDH36qhuryf3ZQmOTKC6tnipPaBuO50kcbktqxO5XSg59G1PAW2IBz14e93umgSIlJJDLimPWCxms0huOM@V4qDEYnDmc1FwkJkoBl4UvS0fftEXB6GEugceCfFckcUSMZOJ/5H7jFBRamkC8FE7tBvnLXlJnV9C0z88LVsO9vf7QfeRlhVGeiCDz9KUtinV3R8 "C# (Visual C# Interactive Compiler) – Try It Online"

HUGE credit to @ASCIIOnly for golfing 30+ ;) At first we were both posting updates simultaneously, but at some point he clearly went to town!

Less golfed code...

    // s is the input as a string
    s=>{
      // t is another string used
      // to hold intermediate results
      var t=s;
      // the algorithm repeatedly
      // processes s and saves the
      // result to t
      do{
        // copy the last result to s
        // and blank out t
        s=t;
        t="";
        // iterate over s
        for(int i=0;i<s.Length;)
          // append either 1 or 2 times
          // the current digit to t
          t+=(s[i]-48)*
            // compare the current digit
            // to the next digit. to prevent
            // an out-of-bounds exception,
            // append a 0 to s which either
            // gets ignored or collapses
            // to 0
            (s[i++]!=(s+0)[i]
              // if they are different, then
              // the multiplier is 1
              ?1
              // if they are the same, then
              // the multiplier is 2, and we
              // have to increment i
              :2*++i/i);
      }
      // continue this until the input
      // yields the output
      while(t!=s);
      return t;
    }