As others have said, the sequence is infinite, though unknowable. Even if math can't say what the precise value is, it can still prove things about the shape of the sequence.

As a simple example, we know that BB(1000) <= BB(1001) (since you can encode the longest running 1000-TM with an extra, unused state as a 1001-TM) even though we will never know either of those values.

The argument that it grows faster than any computable function is similar in that it proves things without knowing concrete numbers. (I'd have to look up the details of the proof to say more)