Hmm, I think you might be a bit underestimating?

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.