Physics and mathematical hyperoperations above exponentiation

[Jenab2]

TL;DR Summary

The question is whether any physical process requires tetration, pentation, or any hyperoperation above the level of exponentiation, in order to be modeled.

I personally don't know of any physical process that can't be modeled without need for tetration or higher order hyperoperations. In my own experience, exponentiation suffices. Does anyone else know about a physical process that can't be well-modeled unless tetration is used in the math?

 

[jambaugh]

I seriously doubt any significant physics will require anything beyond exponentiation. More and more the language of physics has centered around group structure, specifically the Lie groups representing symmetries or dynamical transformations. These utilize the exponential map to translate the parameters (time Lie algebra generators) to the group actions.

Periodic behavior and exponential growth/decay can both be seen to arise within this group context.

This emerges, I believe, because we work in a paradigm of actions on systems which is fundamentally associative (it is built into the semantics of composition of actions). As such they can be iterated in a group structure and that iteration is parameterized by an exponential map. Iterating the actual power operation is horribly non-associative (a^b)^c != a^(b^c) and so doesn't reflect the kind of thing we iterate as an action.

While I've played with some possible application of non-associative product structures in theoretical mathematics (exotic particle statistics) it would be a stretch to find an application of e.g. tetration there. Maybe not impossible though, some wild combinatorics application?