The most mind-bending thing to me about the Busy Beaver problem is that once N gets big enough (probably into the low hundreds of states) the machines can start to examine all of mathematics itself, and so likely are posing extremely interesting questions like whether a particular set of finite axioms is complete and consistent, and checking all possible proofs until finally discovering a gigantic theory smaller than PA or ZFC but still far larger than any finite systems we can ever imagine, and halting once all the possible proofs are enumerated. Depending on how our universe turns out to work on a fundamental level, busy beavers may contain a record of the evolution of our universe (and all the alternatives from different initial conditions) from the big bang until the heat death as a trivial lemma in a larger proof. They'll be full of proofs or disproofs of mathematical conjectures in fancy formal systems we'll never have time to discover even with the infinite axiom schemas available to us.

At least that's the most complex, longest-lasting and most bits-written-to-tape task I can imagine specifying in a few hundred bits. The neat thing is that BB(N) considers all the interesting and answerable questions anyone could ever think to ask with N states worth of bits. Sure, maybe there's some trivial rote task that is truly the biggest for a given N (which I doubt; it seems like rote tasks would fall out of an exploration of a larger conceptual space) but some of its neighbors will be absolutely amazing.