There are plenty of unknown numbers :) For example, the number that is 1 if the Goldbach Conjecture is true, and 0 if it is false.

(BTW, if you want to know the difference between constructive and nonconstructive mathematics, it's that constructive mathematics doesn't let you say "well, it's definitely 0 or 1" because it leaves open the possibility that there's no proof of either termination or nontermination in the current theory... which is basically the philosophical distinction under discussion).

In the book "When Einstein Walked with Gödel" by Jim Holt, It seems like that assumption is not as clear as it seems. I am not able to articulate it well enough and I don't have the book at hand.

If my recall is accurate then there is a bit of a question if there exists a "final number".

There is though, what’s the prime number after 314159265378979323846264338327950288419716939937510582090494459?