Its been a while since my undergrad math, and I couldn't find this by obvious searching.

So I know about hyper operations: addition, multiplication, exponentiation, tetration, pentation etc. And about the various ways to write them.

Now, multiplication is repeated addition. And a summation expression is also repeated addition (just with the possibility of different summands), similarly a product expression is a generalisation of exponentiation with different products. What is the equivalent of tetration?

In other words, how would you write

$\displaystyle a_1^{a_2^{a_3^{...^{a_n}}} $

where $\displaystyle a_1,a_2,a_3, ..., a_n$ are given by some expression in terms of n, and what is it called (if anything), if I want to find out more about its mathematical properties?

The best I came up with was using Knuth's big-arrow notation as:

$\displaystyle \mathop{\mbox{\Large $\uparrow$}^0}\limits_{1}^{n} a_n$

Which would obviously be elegant for things higher than repeated exponentiation, but that zero superscript looks inelegant.

And besides, it doesn't help me know if it has a name or where to find its properties.

Or yet another, simpler, way: what's next in the series: Sum, Product, ...?