Real numbers can have infinite digits, but only after the point, such as pi. The size of any number, even if it's real, can't be infinitely big (a real is between two integer numbers, and integer numbers have to be finite). So yes, 1000(...)00000 with infinite 0s doesn't exist as a number.
Nice conversation anyway.
Yep, that's new to me. So one followed by infinite zeros is not a rational number number. I have read otherwise, but I assume it was a sloppy definition of infinity that works when your not talking about countable numbers.