If we have proof that tree(3) is a number, then we can say that tree(3)+1 is a number that is bigger than tree(3) while also being a number itself. Now that we have proof that tree(3)+1 is a number we can say that (tree(3)+1)+1 has to be a number while also being bigger than tree(3)+1.

You can’t say that it’s the biggest number we have proof about, that’s not how it works. It is however the biggest number used in a mathematical paper, and the biggest number to have been specifically identified