The Professor's base building number converter
Mathematica (REPL environment), 858 total bytes

Here is the 86-byte code snippet for the digit 9:

1;ValueQ@a||(a=0;b=3);a=10a+9;b++;FromDigits[a~IntegerDigits~MixedRadix@Range[b,1,-1]]


The code snippets for the digits 1 through 8 are identical, except that the 9 is replaced by the appropriate digit. The code snippet for the digit 0 is identical, except that +9 is simply deleted.

a~IntegerDigits~MixedRadix@Range[b,1,-1] computes the list of the factorial-number-system digits of a, as long as b is at least as large as the number of digits; FromDigits converts that list of digits to a regular base-10 integer for the purposes of output. (If any of the list elements exceeds 9, something funny happens.)

In Mathematica's REPL environment, computations can be terminated with semicolons to suppress the output; so only the last output in a semicolon-separated chain will be displayed. We recursively define the integer a designated by the snippets, and also a bound b on the number of factorial-system digits needed. The command ValueQ@a||(a=0;b=3) initializes these variables if they're uninitialized (i.e., in the first snippet) and leaves them alone otherwise; then a=10a+9;b++ carries out the recursion. Finally, the initial 1; is for gluing the snippets together: it multiplies intermediate computations by 1 (which we never see anyway).