The Universe is a hierarchy of orders; quarks and gluons, atoms, molecules, cells, multi-cellular and on. So hyperoperators form a natural hierarchy to model multilevel systems. Feynman's Path Integral with the integral removed is just tetration. So I believe the higher operators through self organization/renormalization are directly associated with specific levels of reality, but except for QFT I have no suspicion as to which hyperoperators might be associated with levels of physics. Here is how it might work. Hexation might model chemistry while heptation could model simple biological systems. Thanks for asking. This is fun stuff to work on.
Writing Mathematica software with Stephen Wolfram's support to extend the hyperoperators beyond exponentiation - tetration, pentation and so on, from the natural numbers to complex numbers and even matrices. I do this by extending the iteration of any smooth function to real and complex iterates. http://iteratedfunctions.com/ and http://tetration.org/.
Physics has two mathematical methods for it's representation, partial differential equations and iterated functions. My work is more general than physical systems or even the universe because I can consider both measure and non-measure preserving systems. I am looking at AI applications as a system that is tuned to solve for physically possible models.