Are there any integers that are considered uncomputable based on their extremely large size alone?

Question

This is difficult to frame, but my question is if there are any finite integers that are so large that they are uncomputable in principle. Meaning that they cannot be handled by mathematics even in theory. So is there some kind of limit in abstract math where a positive number is so large that it is no longer considered encodable by symbols, even ignoring the lack of resources in the physical world to write it down. Like a fuzzy boundary where analysis isn't really sufficient to talk about them. Or are all finite numbers ultimately considered within the realm of consideration at least in an abstract sense?

Answer 1

Every integer is computable in the technical sense, because an integer is ultimately just some finite string of digits and so there always exists a program whose code simply consists of "print (string of digits)."

If the integer is extremely large, then this program will also be extremely large, and this is mostly unavoidable by a straightforward counting argument which establishes that most strings have the largest possible Kolmogorov complexity. But ignoring the lack of the necessary resources to physically write down such extremely large programs, this isn't an issue for the technical definition of a computable number.

The discussion here may also be helpful.