1. Where can we find prior work on the subject?
The model of exponential fields seems to be related, but these are defined with an exponential function instead of a commutative operator, and no inverse function. Furthermore, an exponential field defines a single field structure, while $\mathbb{H}_n$ defines $n-1$ field structures on the same object, alongside $n$ group structures. Therefore, exponential fields and hypertypes are quite different.
Many of the ideas outlined above where first proposed by @goblin back in 2013, but only in the context of $\mathbb{R}$, while distributivity was questioned but not established. One of the main benefits of the Hypertype Theory prposed here is to provide a recursive sequence of larger and larger objects that are isomorphic to subsets of $\mathbb{R}$ while providing recursive group and field structures.
5. Can $\pi$ be naturally introduced as a term of $\mathbb{H}_n$?
If my understanding of Schanuel’s conjecture is correct and if the conjecture is true, then $\pi$ is not a term of $\mathbb{H}_n$.
7. Is this interesting?
This is interesting for several reasons:
First, it provides a recursive set of objects $\mathbb{H_n}$ that have very simple definitions yet unique properties. Among them, the existence of $n$ group structures and $n-1$ field structures. And because of the recursive nature of the construction, many properties exhibited at level $n$ become true at any level. This provides a very powerful model to work with.
Second, the construction centered around $e$ introduces this transcendental number in a very natural manner. This is important, because $e$ is one of the few common numbers that is believed not to be a period. Therefore, the combination of Hypertype Theory and Period Theory provides a solid framework for the classification of transcendental numbers.
Third, the adoption of a commutative power law sheds a new light on the notion of power laws and exponential operators, which are traditionally non-commutative. The introduction of a different viewpoint on a fundamental operation might foster innovation or lead to some interesting discoveries.