Graham's number is typically cited as the largest number that has ever been seriously used in a mathematical proof, as the Wikipedia article says, but Busy Beaver passes it very quickly.
One of the nasty little characteristics of BB that Aaronson describes, but can stand to be reiterated since I've noticed many people don't seem to quite follow it, is that any mathematical algorithm you can encode into a Turing Machine in some number of rules X, is therefore at most the top-end bound of BB(X), and more likely (which in this case is serving as mathematician's-understatement-speak for "certainly"), not even close. And for all that the TM "programming language" isn't that efficient, certainly almost anything you can describe in a reasonable English paragraph is, say, well less than 100 states in a TM. For all of Graham's Number's stonking size, in algorithmic terms, it still isn't that complicated.
Generally speaking, human attempts to create Big Numbers by fancy recursively-applied algorithms are still quite simple to put into a TM. This is why it's an interesting point that we don't even understand what exactly a BB(5) solution is doing; a mere five states and the human mind is already lost, lost, lost. I think of Busy Beaver as showing in its own way just how thoroughly we have not mastered math, and never will master more than a tiny little island that happens to be relatively easy to follow, because BB shows us that we don't have to swim out into the great sea of "all math" very far at all for there to be monsters.
Whether or not BB(n) is even defined is an interesting philosophical question.
If n is a modestly large number, it can be proven that in no consistent axiom system below a certain size can any explicit number written out in base 10 EVER be proven to be an upper bound for BB(n). If we make n something like 100 million, that encompasses all possible axiom systems that are human comprehensible.
A "finite" number with no provable upper limit - how much sense does it make to say that is well-defined? It is enough to make you take up constructivism! (Or quit math for something more sensible. Like politics.)
One of the nasty little characteristics of BB that Aaronson describes, but can stand to be reiterated since I've noticed many people don't seem to quite follow it, is that any mathematical algorithm you can encode into a Turing Machine in some number of rules X, is therefore at most the top-end bound of BB(X), and more likely (which in this case is serving as mathematician's-understatement-speak for "certainly"), not even close. And for all that the TM "programming language" isn't that efficient, certainly almost anything you can describe in a reasonable English paragraph is, say, well less than 100 states in a TM. For all of Graham's Number's stonking size, in algorithmic terms, it still isn't that complicated.
Generally speaking, human attempts to create Big Numbers by fancy recursively-applied algorithms are still quite simple to put into a TM. This is why it's an interesting point that we don't even understand what exactly a BB(5) solution is doing; a mere five states and the human mind is already lost, lost, lost. I think of Busy Beaver as showing in its own way just how thoroughly we have not mastered math, and never will master more than a tiny little island that happens to be relatively easy to follow, because BB shows us that we don't have to swim out into the great sea of "all math" very far at all for there to be monsters.