Why do we stop at exponentiation stage in arithmetic of natural numbers? In natural numbers the unary successor operator $S$ is the most natural function which maps each number to the next one. Furthermore we may consider the binary relation $+$ as an iteration of $S$. Also $\times$ is an iteration of $+$ and $\exp$ is an iteration of $\times$  i.e.

$\forall m,n\geq 1$
$m+n:=\underbrace{S(S(S(\cdots S}_{n\ \text{times}}(m))))$
$m\times n:=\underbrace{m+m+m+\cdots+m}_{n\ \text{times}}$
$m^n:=\underbrace{m\times m\times m\times \cdots\times m}_{n\ \text{times}}$

We build the rich arithmetic of natural numbers via above three natural operators. Also they show many complicated mutual relations with each other. But, why do we stop here in arithmetic of natural numbers and don't go forward with continuing iterating operators again and again? i.e.

$m*n:=\underbrace{m^{m^{m^{.^{.^{.^{m}}}}}}}_{n\ \text{times}}$
$m\circledast n:=\underbrace{m*m*m*\cdots*m}_{n\ \text{times}}$
$m\circledcirc n:=\underbrace{m\circledast m\circledast m\circledast \cdots\circledast m}_{n\ \text{times}}$
$\vdots$

The point is that maybe there are rich interactions between these new natural operators and the ordinary arithmetic operators of natural numbers. These interactions may unfold some deep aspects of long standing open questions of number theory which hopefully can lead us to a solution itself.

Question: Why do we stop at exponentiation stage in arithmetic of natural numbers? Is there any mathematical or philosophical problem about defining such generalized operators and working with them as well as successor, sum, multiplication and exponentiation? Are these "unnatural" in any sense? If yes, what does this "unnatural" essence mean?  Did these extended set of operators on natural numbers appeared in any text before? If yes, please introduce your references.



Why do we stop at the exponentiation stage in the arithmetic of natural numbers?

We don't actually stop there. Knuth's up-arrow notation generalizes the operations up to any level you want and Conway's chained arrow notation even further. Note though that all these generalizations are applicable only for $n\in\mathbb{N}$.
Inability to generalize and hence stop at exponentiation, is a problem only when you try to extend the nomenclature to include definitions for REAL $n$. That's an entirely different problem and opens a rather big can of worms. See tetration on Wiki for details.
