I think you’re confusing “arbitrarily large” with “infinitely large”. See Wikipedia Arbitrarily large vs. (…) infinitely large

Furthermore, “arbitrarily large” also does not mean “infinitely large”. For example, although prime numbers can be arbitrarily large, an infinitely large prime number does not exist—since all prime numbers (as well as all other integers) are finite.

For integers I disagree (but I’m not a mathematician). The set of integers with infinite digits is the empty set, so AFAIK, it has probability 0.