Digit Product Sequences
JavaScript (ES6), 81 67 bytes
Saved 1 byte thanks to @l4m2

f=(n,x=1)=&gt;x&lt;n?f(x,n):x&gt;n?f(+[...n+''].reduce((p,i)=&gt;p*i||p)+n,x):n

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Commented
f = (n,                   // n = current value for the 1st sequence, initialized to input
        x = 1) =&gt;         // x = current value for the 2nd sequence, initialized to 1
  x &lt; n ?                 // if x is less than n:
    f(x, n)               //   swap the sequences by doing a recursive call to f(x, n)
  :                       // else:
    x &gt; n ?               //   if x is greater than n:
      f(                  //     do a recursive call with the next term of the 1st sequence:
        +[...n + '']      //       coerce n to a string and split it
        .reduce((p, i) =&gt; //       for each digit i in n:
          p * i || p      //         multiply p by i, or let p unchanged if i is zero
        ) + n,            //       end of reduce(); add n to the result
        x                 //       let x unchanged
      )                   //     end of recursive call
    :                     //   else:
      n                   //     return n