Most numbers are uncomputable because most real numbers are infinite strings of non-repeating digits, which obviously would never fit in a computer. More interesting non-computable numbers include the Busy Beaver problem https://en.wikipedia.org/wiki/Busy_beaver There doesn't seem to be a way to compute these numbers without basically trying out every possible answer.
But is the universe even modeled by real numbers or is this solely a concept of the mathematical domain?
Hawking calculated the Bekenstein bound, which proved an upper bound on the amount of information that can be contained within a given volume. As black holes are maximal entropy objects, one can compute the entropy for a certain black hole volume and equate that with Shannon entropy to determine the number of bits. This suggests, to some degree, that the fundamental particle is the bit.
(This also proves that a true Turing machine with unbounded memory is not physically possible.)
This whole thing is in the "mathematical domain". The point is that you can define a number in such a way that it clearly exists, but there's no good way to calculate it.
