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

where 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:

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, ...?