I think you're mistaking the map for the territory. A googolplex is a representation of a number, but not the number itself, although it's simple enough that we can get away with using the representation as it's obvious what the form of the number would be. However a number such as tree(3) is unimaginably bigger, but more crucially, we don't know anything about the form of the number beyond its size and we can't sensibly use it in calculations.

Now both of those numbers are finite and we could try to figure out how many numbers we could "describe" such as tree(3), but that would be limited by the number of symbols (i.e numbers, operators, letters and words) that could be used (i.e we would have less than a googolplex different numbers that could be represented using maths, language and thought). That's still going to be a finite number.