What's the largest number?
The construction of the nat-nums you alude too relies on infinity as an axiom, so the construction itself doesn't prove jack.
For all practical intents and purposes "infinity", in a very narrow sense, is indistinguishable from arbitrary large numbers.
Your rhethoric question leads to contradiction: By the same line of reasoning, if I don't know what the weather will be next year, that must mean it wont exist, so time is finite, which contradicts your implications.
(Oh no, am I a troll? I think I'm a troll now.)
The whole point of having no upset bound on numbers is that this kind of game can be continued arbitrarily.
But I think you can seriously make the argument that not only is "largest number" an empirical question, but the answer depends on your units. I'm not sure that's more absurd than the stuff in TFA.
However, what if we want to theorise about what, if anything, sits outside the observable universe? You seem to be suggesting that it will always be finite, which is optimistic to say the least. Just because we can't observe something doesn't mean that it isn't real, and that we won't get utility out of modeling it. Infinity might be out there.
I hope you didn’t want a number you could actually figure out....
The largest number I saw written out is the number of symmetries in the monster group, spelled out on wikipedia (iirc).