So this turns out to be less interesting mathematically than you might think, because the entire thing about Busy Beaver numbers is that it is using the full power of computation available to it for a given number of states, so as soon as the "normal" Busy Beaver reaches a number of states sufficient to describe whatever trick you're interested in pulling, the Busy Beaver sequence itself already uses that trick!

One of the most important sentences in understanding them is one that's easy to pass by in the original work: "Indeed, already the top five and six-rule contenders elude us: we can’t explain how they ‘work’ in human terms."

That is, while we humans are thinking we're all clever by defining repeated iterations of the BB ruleset itself, all of these things that we think we are being so clever with are actually very very compactly described by Turing Machine states. Meanwhile, even by BB(6) the TM is metaphorically "using" incomprehensible tricks to be larger than we'd even dream. If repeated iterations of BB'ness can be trivially described in, say, 10 states, and let's say we reserve 4 states for BB(4) to describe the number of times to iterate, our hand-constructed "clever" re-iteration of the BB procedure will be blown away by some other 10-state machine that we can't even imagine would ever terminate.

So on the one hand, yes, BB(BB(20)) is incomprehensibly larger than merely BB(20), and yet, on the other hand, in a profound way, it also doesn't matter. Busy Beaver in a sense shows us numbers that have all the practical properties of infinity, even if they are not technically infinite. In a sense BB(30) < BB(31) is obviously true, yet, ultimately, a humanly meaningless statement, since grasping either number in any sense beyond its mere definition is impossible. We might as well say that "blibble" is less than "bloo". And not merely impossible in the sense that it is impossible to even really properly grasp just how far it is to Alpha Centauri, but impossible in the sense that we are literally incapable of even constructing mathematical tools that will let us grab on to and manipulate such numbers... deeply, profoundly, computationally impossible for us to understand, not merely "mind-blowing", but impossible for us to understand in the absolute strongest sense of the term.

Similarly for trying to catch up to BB with yet more iterations of exponentiation... the procedure we humans use is shockingly, shockingly simple to describe programmatically, which means that BB automatically already encompasses practically all such tricks very early in its sequence, which means you can't defeat it that way.

Busy Beaver is metaphorically (again) much smarter than you, and it's not worth your time to try to outsmart it to make yet again bigger numbers.

This also, probably correctly, implies that attempting to adjudicate some contest in which some people write BB(BB(BB(x)))) vs some other BB-based notation is also impossible and you'd actually fail out of the contest for writing an ill-defined number, as if we can't compare the two numbers for which is larger, even in principle, it is for the purposes of this contest, ill-defined. Busy Beaver in a sense also sets the limit of what a well-defined number even can be, and is thus in some sense the natural limit of this contest by virtue of simply transcending through sheer incomprehensible size all our computationally-naive human tricks for making big numbers, or even just numbers of any size at all.

It's really a profound sequence.