Comment by xelxebar

Busy Beaver gets a lot of love, but the fast growing hierarchy is both constructive and can go way, way, waaaaay beyond current known BB bounds. This makes their size much more viscerally apparent than gesturing vaguely at BB(BB(BB(100))) or whatever, IMHO.

David Metzler has this really cool playlist "Ridiculously Huge Numbers" that digs into the details in an accessible way:

https://www.youtube.com/playlist?list=PL3A50BB9C34AB36B3

By the end, you're thinking about functions that grow so fast TREE is utterly insignificant. Surprisingly, getting there just needs a small bit of machinery beyond Peano Arithmetic [0].

Then you can ponder doing all that but making a tiny tweak by replacing succesorship with BB. Holy cow...

[0]:https://en.wikipedia.org/wiki/Theories_of_iterated_inductive...