The magnitude of 5^^^^5 is itself mind-bogglingly huge.

I assume you want to know how many digits it has? Let's start with 5^^^5. Log10 5 = 0.69897, so 5^^^5 = 55^^5 = 100.69897(5^^5). So the number of digits is 0.69897(5^^5) + 1. You can forget about the +1, because this comes out to 2.0831e+17. That's not the number, remember, but the number of digits it has - about 200 quadrillion. (Considering that our puny concept of "digits" is not at all sufficient to express numbers of the magnitude that we're dealing with, I don't think it affects it too adversely that I'm cutting my answers down to one significant figure.)

In fact, this number doesn't quite bring us into the "too huge for words" range, because it is significantly less than a googolplex, which has 1e+100 digits. (Yes, I can just hear some wise guy calling out "plus one!" right about now.) But now we come to 5^^^^5, which is what was asked about in the first place. This is where it gets mind-boggling. I can't tell you how many digits the number has. There are somewhere around 100 quadrillion digits in the number of digits of the number! If you don't have a problem with that, you probably aren't thinking about it right.

Consider a sequence that goes 1, 10, 10000000000,
100000000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000000000000, etc. Notice how fast it's growing? 5^^^^5 would be around the hundred-quadrillionth (give or take an order of magnitude) term of this sequence.

You can go take some aspirin now.