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?
Gap between $BB(n)$ and second largest $BB(n)$
computability