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.
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!
Is there a laymans explanation for how you could apply this in an 'ordinary' setting, in other words is this from my non-mathematical background a useful construct or should I treat it as a curiosity?
I find it interesting that there is a simple order to the operations +, *, ^ and so on that apparently can be extended further than I was aware of, is there a limit to that?
How would a fifth element in that series be applied?
Edit: as to the last, it seems the series builds on the previous operation in the series to build the next, so possibly 'pentation' would be a series of tetrations?
2+4 = 6
2x4 = 2+2+2+2 = 8 (edit2: using an 'x' here because hn eats asterisks thinking you mean italics)
2^4 = 4x4x4x4 = 256
2^^4 = 2^(2^(2^2)) = 65536
2^^^4 = ...?