Major references

[MphLee]

Oh, I apologize Mphlee. The reason I consider your list "not very analytic"--is because there are no analytic papers on these subjects.

idk man, u don't leave me fully convinced. OFC there are not analytic rigorous papers on hyperops and tetration at the moment. ofc that makes you claim trivially true. But in my iteration list, that is not complete for sure, there are items that I can't imagine how you could call non-hard-analytic. There is also Milnor there.
Try to check that first list better when you have time. Maybe there are some gems that you can use.

There really are ZERO hard well developed papers on the hyperoperators, as analytic objects. The closest you'll find is Kouznetsov, who definitely leaves things to be desired (Just because his calculator works, and his intuition/ad hoc reasoning is very correct; doesn't mean it's rigorous).

So, I think I misspoke a tad. What I mean is that there are no good analytic papers on these subjects, and the few you will find, are severely lacking in rigor. Even myself, I only consider 80-90% of my results to be "proven rigorously"--but then, they've never truly been vetted, other than my calculations--which brings us back to Kouznetsov's level of "truth".

Ok Agree.

Where as the good papers, that I identified, are very... not sure the word, "foundational," "about the structure of hyper-operators". Which definitely lead to a "categorical" understanding. I'm aware you're probably the most eminent person on this planet on "Hyper-operators & category theory", because no ones ever touched this before. (No one cares about these subjects, and I love it, because it allows me to work without fear of rediscovering some 100 year old formula no one cares about --doing this, at least it's a new formula... no one cares about).

My main grievance, with saying "it's not a very analytic list"--is, I guess, it's not really a very "RIGOROUS ANALYTIC" list. If anything, a lot of work in tetration/hyper-operation/nested function theory tends to be numerical based. So I would call it more so, numerical analytic. And much of the results end up being effective calculators; but not rigorously proven constructs.

...

So I struggle to call much of hyper-operation/tetration/iteration theory actual analysis. When usually, at best, it is Numerical Analysis. This would definitely fall under Kouznetsov's work. Where the exception is Kouznetsov & Trappman, which is absolutely Analysis in the hard rigorous sense.

For example, I'd like to have a quick opinion from you, maybe in a separate thread, on
-1936 Morgan Ward, F. B. Fuller - The continuous iteration of real functions"
-1956 - Erdos, Jabotinsky - On analytic iteration
-1999 Belitskii, Lyubich - Abel equation and total solvability of linear functional equations
-2001 Carracedo, Alix - The Theory of Fractional Powers of Operators
-2003 Keen, Lakic - Forward iterated Function Systems

In particular Linda Keen (http://comet.lehman.cuny.edu/keenl/publications.html ) seems to have continued to build lot on the theory of Gill. Do your work fully subsume it, were you aware of them?