This (http://en.wikipedia.org/wiki/Knuths_up-arrow_notation#Introd...) describes how it can be extended without limit. And there's are numerical examples at the end of that section demonstrating how. And this (http://en.wikipedia.org/wiki/Pentation) focusses specifically on the next step on "pentation".

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.