Comment by stillsut

Is there some N such that BB(N) can produce any "arbitrarily large" number? Not infinity because that would mean it doesn't halt, but that there is no number which can not be exceeded by saying in effect "plus 1".

Or, as N becomes large and can emulate the english language use berry's paradox style algorithm to show there is no limit to what size you want.

Another approach would be to use a probability-based event for halting, and so it is always possible that a series of coin flips could keep coming up heads 1,000,000 times in a row, or a zillion times, see: st. petersburg paradox.