Oh, wow! People have managed to find the number BB(5) (!)
https://discuss.bbchallenge.org/t/july-2nd-2024-we-have-proved-bb-5-47-176-870/237
It's 47 176 870, and this has been verified in Rocq [née Coq], and *that* formal proof has been scrutinised by experts in that language to check that it really does do what it says on the tin.
Incredible.