But does "a computable function that grows faster than BB(N)" actually mean that f(N) is always greater than BB(N), or could it be taken to simply mean that f(N) > BB(N) for a proportion of N that trends towards 1? In other words, could you have a function that almost always upper bounds BB(N), except for a tiny (but still technically infinite) proportion of exceptions? Such a function would be larger than BB in a meaningful sense, but if the sequence of exceptions is not computable, it wouldn't be possible to solve the halting problem with it. There may be a problem with this concept, but it doesn't appear obvious to me.
I am curious if there is a proof for the less obvious case, though. Could a function be greater than BB most of the time? What are the conditions, exactly?
