[amelius]

I was wondering about something. If I have the sum x+x+x...+x (n times), then that is the same as x * n. If I have the product x * x * x ... * x (n times) then that is the same as x ^ n.

What is it called when this is generalized? E.g. call + op1, call * op2, call ^ op3. What would op0 be? And what would op0.5 be?

How does the unit element for these operations behave?

And the rules for associativity, commutativity, for increasing order of the operation?

[sa1]

It's called a hyperoperation: https://en.wikipedia.org/wiki/Hyperoperation, and it's defined only for natural numbers, so op0.5 would still be undefined.

[n4r9]

The generalisation that you're hinting at is known as Knuth's up-arrow notation [0].

There's a number known as Graham's number [1] which is defined in terms of up-arrow notation and was for a while the largest specific positive integer to have been used in a mathematical proof.

[0] https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation

[1] https://en.wikipedia.org/wiki/Graham%27s_number

[soVeryTired]

The operation beyond exponentiation is known as tetration: towers of exponentials. It's not associative, and therefore it's difficult to work with and hasn't received much interest from the mathematical community at large.

This might sound harsh, but unfortunately it does tend to attract 'cranks'. I think the reason for this is that there's a clear pattern (as you picked up on), that doesn't require formal mathematical training to spot. Amateurs get excited about the prospect of discovering something `new', without realising how hard it is to say anything deep about the topic.