Brouwer distinguished between actual infinity and potential infinity, with the former inadmissible in intuitionism. He would say “the largest natural number” is a meaningless statement. You could only have the “largest so far”.
Intuitionism invites you to construct as large of a natural number as you want. It simply does not admit “the complete set of all natural numbers”.
The point of intuitionism is the idea that all mathematics is constructed by human creativity and logic. Brouwer rejected the idea that mathematics was “already out there, waiting to be discovered”. This is a subtle point. It says that mathematical objects do not exist in some mind-independent realm of truth. It has the nice property of avoiding some of the problems of classical mathematics that Hilbert got caught up in.
“So far” with respect to some particular problem you’re solving or process you’re following. If you’re talking about all problems, you’d have to do some research. Perhaps OEIS [1] could help you. Or you could interview some number theorists to find out who is working in large numbers and ask them.
Asking the question in general, though, is equivalent to asking “what is the largest number anyone has thought of?” I would have to interrogate your motives to know why you would want to ask such a question.
Intuitionists would scoff at the idea of a “number larger than anyone could describe.” That’s heading into the territory of the interesting number paradox [2]. Intuitionism avoids this paradox by stating that numbers (and sets of numbers) have no existence independent of their construction.
" “what is the largest number anyone has thought of?” I would have to interrogate your motives to know why you would want to ask such a question..."
I ask because it's absurd. A particular large number is a logical thing regardless of whether a person thought of it before or whether it describes a distinct natural phenomenon. This follows from the set theory axioms, as well as Real numbers and infinitely many other number systems. To restrict our capacity to think of numbers to only numbers that people can enumerate or numbers that describe natural phenomena seems arbitrary and onerous. We construct the Real number simply by considering the limits of convergent sequences to be numbers. I don't see anything transgressive about this idea. Sure, some numbers are not computable, and other numbers can't be represented with marbles; they still have utility and there is no reason to make the concept of "numbers" exclusive.
