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I didn't see this question in here but it was asked in quora and it was interesting to me that no one had any satisfying answers. some people suggested that it's because exponentiation describes the solutions to some differential equations. but isn't that part of the convenience? differential equations are talking about differences that naturally* occur till exponentiation this is why they are related. why arithmetic in our world reduces to exponentiation? why everything (well most of it) conveniently packs up till exponentiation?

*: exponentiation provides solutions because rates of change in nature and the practical numbers are conveniently packed in hyperoperations with n<4.

John Conner
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I'm posting this as an answer, but it's really somewhere between an answer, reasonably informed speculation, and an opinion.

Let's start with an assumption: if there were no upper bound on the level of hyperoperations common in nature, then the result would be a badly disordered universe, which isn't amenable to life. So given that the universe doesn't seem to be disordered, it was already very likely that there would be some upper bound.

Now, math was originally developed specifically for the purpose of describing the world. Addition was invented largely to describe combining collections; multiplication for calculating areas. If we lived in a one-dimensional world, it's not unreasonable to suppose that multiplication wouldn't have been invented. In other words, we developed "standard" operations to match the phenomena we saw.

The conclusion here is that, if tetration were common in nature, we would already consider it just as "convenient" as exponentiation. In other words, it's not a coincidence that nature happens to stop at the same point we do - it's that we stopped making new operations when nature did.

Some mathematical observations, as well:

  • Most operations that show up frequently in nature have simple domains; many are defined on all real numbers, and those that aren't (like $\sqrt{x}$) are defined on a simply-connected set. But tetration isn't like that; there's no reasonable way to define $^{-2}2$, for example, because (since $^02 = 1$) $^{-1}2 = 0$, and as a consequence you can't define $^{-n}2$ for any integer $n > 2$; but if $^{-1/2}2$ is defined to be anything other than $0$ or $1$, $^{-5/2}2$ does have a reasonable definition.
  • Exponentiation is also the first hyper-operation that is not associative, so tetration involves arbitrarily picking a particular ordering; why would 2^2^2 mean $2^{2^2}$ and not $(2^2)^2$? That suggests that the idea of tetration is fundamentally less natural than exponentiation.
Reese Johnston
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    Possibly related to that last point is the idea that $A+B$, $A\times B$, $A^B$ all have a combinatorial meaning when $A$ and $B$ are arbitrary sets (respectively disjoint union, Cartesian product, function set), but set tetration ${}^AB$ only seems to have a combinatorial meaning (e.g. the set of hereditary $B$-names of rank lower than $A$) when $A$ is an ordinal, and thus higher order hyperoperations only make sense when both $A, B$ are ordinals. This highly restricts the cases where tetration, pentation, etc. can appear as a solution of any natural counting problem. – pregunton Jan 19 '20 at 19:13
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The closest I have ever found is Feynman Path Integral with the integration step removed.