There is of course not currently any upper bound on BB(6).
I find the question about a probvious upper bound more interesting - There is also not a probvious upper bound on BB(6), as this would require at least some understanding of the high-level behavior of all remaining holdout machines. However, there may soon be a 'probvious' upper bound on the value of BB(3,3) (BB for 3-state, 3-symbol machines). Up to equivalence, there are four remaining machines to decide to find the value of BB(3,3). One is a 'probviously halting' machine which will be the new champion if it halts, and for which probabilistic models of its behavior predict with high probability an exact halting time. One is a 'probviously nonhalting' machine. The two other machines are not well-understood enough to say whether they have any probvious long term behavior, but some suspect that they both 'probviously nonhalt'. If this is true it could be said that a 'probvious' upper bound exists for BB(3,3).
Very many - I think the number is several thousand. Several sporadic 6-state machines have been solved, though, and there are currently about 2400(?) unsolved machines. Among these are several Cryptids, machines whose halting problem is known to be mathematically hard
I find the question about a probvious upper bound more interesting - There is also not a probvious upper bound on BB(6), as this would require at least some understanding of the high-level behavior of all remaining holdout machines. However, there may soon be a 'probvious' upper bound on the value of BB(3,3) (BB for 3-state, 3-symbol machines). Up to equivalence, there are four remaining machines to decide to find the value of BB(3,3). One is a 'probviously halting' machine which will be the new champion if it halts, and for which probabilistic models of its behavior predict with high probability an exact halting time. One is a 'probviously nonhalting' machine. The two other machines are not well-understood enough to say whether they have any probvious long term behavior, but some suspect that they both 'probviously nonhalt'. If this is true it could be said that a 'probvious' upper bound exists for BB(3,3).