Is there a general theory about repetition of the same operator?

Question

Is there a theory about repeating the same arithmetic operation? E.G. multiplication is a repetition of addition, and exponentiation is the repetition of multiplication.

Answer 1

My answer is maybe less about giving an answer, and maybe more about giving some perspective. Maths is just convention; numbers don't have a meaning thought you can think about it the way you like.

Actually, are you looking for algebra theory ? It all comes down to "how" you define operators. What you say that all operators can be reduced and it works well, it's perhaps because you are talking about complex numbers, but it might be true that:

$$ a^5 \neq a \times a \times a \times a \times a $$

Take a look at rings if you haven't yet, and fields. Those are sets upon which you define an operator. There is no reason why the addition (or first law) would have anything in common with multiplication (second law). Actually, in group theory, one writes

$$ g^3 = g + g + g $$

if $g$ is an element of the group, and $+$ is the law inside the group. LHS is just a common, useful (short cut) notation.

Answer 2

As Rob Arthan pointed out in the comments, Knuth's up arrow notation is the way to go, and the sequence is called the hyperoperation sequence.