Stax, 9 bytes
ü◘ø⌠Δ>0↔å Port of my MathGolf answer. This is only my second Stax answer, so there might be a shorter alternative.
Try it online or try it online unpacked (10 bytes).
Explanation (of the unpacked version):
0 # Start at 0 w # While true without popping, by using everything else as block: x+ # Add the input-integer c # Duplicate the top of the stack E # Convert it to a list of digits |M # Get the maximum of this list 1> # And check that it's larger than 1 # (after the while, the top of the stack is output implicitly as result)