Addition, multiplication and exponentiation all have pretty clear day to day applications.

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 = ...?