We know that the multiplicative operator is construct by iterating the additive one, and the power operator is construct by iterating the multiplicative one.
Which makes us wonder if we can construct a decent operator, lets name it $\triangle$ (triangle) so that the addition would be an iteration of triangle, i.e. $$\underbrace{a\triangle \cdots \triangle a}_{n \text{ times}}=a+n.$$
May be such a construction would have to take place in $\mathbb R\cup \{-\infty\}$, so that $-\infty$ could be the identity element for $\triangle$. May be not.
May be one can use the natural logarithme or the exponential which can upgrade or downgrade an operator in some way.
Anyway, i just want to know any ideas you may have about that kind of operator.
Have you ever read about Peano Axioms?
The function you are looking for is the successor function $S(n)$. It is a function defined on natural numbers as $S(n):= n+1$. Basically, given a natural number $n$, $S(n)$ outputs the following natural number, often referred to as the successor.
For instance, $1$ is oftend defined to be $S(0)$. Obviously for every $a\in \mathbb N$, $n \in \mathbb N$ we have that
$$a+n=\underbrace{S(S(\dots(S(a))}_{n \text { times}}$$