Zeration

[MphLee]

a °  b =  a + 1  ,    if  a > b
a °  b =  b + 1  ,  if  a < b
a °  b =  a + 2 = b + 2 , if  a = b

Under that definition of zeration, a°-∞ would be equal to max(a,-∞)+1. Not a.

Nope, the definition says nothing about its evaluation at \(-\infty\). That evaluation is derived by another rule.
The main purpose of the Rubtsov-Romerio's approach is to propose a solution that satisfies some constraint, and where it fails to compute to look for more fundamental and natural principles from where we can derive new constraint on the solution.

Their solution to the problem is the original Zeration. I mean that nowadays we use the term zeration for all the 0th ranks operations of a given Hyperoperation family. But the therm Zeration was created original by Rubtsov and Romerio to denote that particular solution.
Note also that their proposed solution, that they extended to a new field \(\mathbb R\subseteq \Delta\), has presents a discontinuity at \(-\infty\), that's why the limit doesn't agree.

The info on this is not well presented in this thread nor in many other threads (except a few where I hope I had given a reasonable review of it). On top of this add that this research around "pre-additions" was always conducted mainly by non professional mathematicians and rejected as trivial/stupid by professional mathematicians. I know it may seem an annoying state of the art but there is lot of hidden folklore around this and the original papers are not easily accessible anymore.
I have much of the material though, I might share some, one day.

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To add to the correct JmsNxn remarks you can see it as follows:
evaluating something defined on the reals at infinity means evaluating a function at something that is outside it's domain. It is impossible by definition.
In order to do that you can extend the concept of evaluation or you can extend the domain.
The first solution is the definition by limit or the analytical route. You define the value at infinity as a limit, extending the meaning of function evaluation.
The second solution is by extending the domain, i.e. the algebraic route. You adjoin new elements to the domain, but then all the previous algebraic rules could fail. You can do this in many non-equivalent ways. For example by forming the hyperreals you lose the archimedean property, a fundamental property for analysis but also you lose the uniqueness of infinity because you have different kind of infinities and infinitesimals.