I was wondering if this had any implications for BB(6) but it seems that can't be written down exactly (not enough room in this universe) so BB(5) is the last one we'll see an exact value for.
My understanding of up-arrow notation is: 2 ^^^ 5 means
2 ^^ (2 ^^ (2 ^^ (2 ^^ 2))) (with five 2's)
= 2 ^^ (2 ^^ (2 ^^ (2 ^ 2)))
= 2 ^^ (2 ^^ (2 ^^ 4))
= 2 ^^ (2 ^^ 65536), since Wikipedia calculates 2 ^^ 4 = 65536.
Now 2 ^^ 65536 means 2 ^ (2 ^ (2 ^ ...))) with 65536 two's. Call that number N (too large to really describe).
Then 2 ^^ N is 2 ^ (2 ^ ( ...)) with N two's.