If $HT(n)$ is the set of halting times of $n$-state Turing machines on a binary alphabet with empty initial tape, then $BB(n) = \max HT(n)$.
What can we say about the second largest number in $HT(n)$? Call this $BB_2(n)$.
$BB_2(n)$ is trivially uncomputable, since it lets one compute $BB(n)$: just wait for one more machine to halt. Naively, I would expect the gap $BB(n) - BB_2(n)$ to be "busy beaver-like", growing faster than any computable function. Is this provable?