hehe, funny, I just caught on to that (see edit above).
It's pretty logical really, but I can't seem to come up with what it should work out to. But the notation, once you go from ^ for exponentiation to ^^ for tetration immediately suggests ^^^ for the next step. I feel pretty dumb now for not realizing right away that was what was intended. (edit: I suck at math, in case that wasn't clear yet...).
Thanks!
I'm not aware of any "real-world" applications of tetration in the way that eg exponentials can describe certain physical or economic processes. But it does have some genuine uses in pure mathematics such as calculating the upper bound of some numbers. It's not just making big numbers for the sake of it.