It is my understanding that numbers & games have as there left & right options sets. I think this means that numbers & games are themselves sets. Gaps on the other hand can have (proper) classes for options. I think this also means that gaps themselves are (proper) classes. I'm curious what might be beyond numbers, games & gaps; additionally what kind of (set / category / type) theory would we need to work in to accomplish going further.
In Going beyond the surreal numbers it is mentioned in a comment that
If you work in a framework which allows talking about proper classes (like NBG or MK theories), then you can extend it one more time. If you have a theory, you can extend it twice. And so on.
Additionally, the answer states
Your proposal is to continue filling cuts after all ordinal birthdays are completed.
All such cuts will have cofinality Ord on one side or the other, and so each such cut will take a proper class to represent it. So the first thing to say is that there will be certain set-theoretic foundational difficulties with undertaking the construction. For example, this is not straight-forwardly a ZFC construction, but you could proceed in GBC for a step or so. To proceed much further, you will need stronger second-order set-theoretic axioms, such as the axiom ETR of elementary transfinite recursion, which allows one to undertake recursions on proper class well-founded relations whose rank exceeds Ord.
There is another question (Is there a “more powerful” form of set theory that would enable this?) which asks
I was wondering about this. In Conway's "On Numbers and Games", he discusses the "surreal numbers", and in one point mentions that they are full of "gaps". That the surreal number line is riddled with gaps. Namely, what he mentions is that these gaps occur for "cuts" between proper classes of surreal numbers, whereas ordinary surreal numbers are cuts between sets.
He then goes and mentions how that we cannot collect these together, it would be an "illegal" (undefined?) object in conventional set theory. Which makes me wonder -- could there exist some greater, more powerful form of set theory that could enable this kind of "higher-order collection" to exist? And then we could talk about the properties of "all surreal number plus all gaps in a single continuum". Or is there a good, fundamental reason that this simply cannot be done? If so, what is it? And if it can be done, what kind of properties would this monster have, anyways?
The answer to that question states a similar sentiment to the previous answer
You can extend ZFC to allow classes and 2-classes (classes of classes), but this makes things complicated and very delicate. Instead a common cure for the problem is to assume there exists a set model of ZFC (e.g. if there is an inaccessible cardinal), and talk about the surreals of that model from an external fashion. Namely the collection of all cuts is a set in the universe, but not a definable collection within that model. Again, this is full of delicate points.
I'm not sure if there are any issues with games & set theory. It definitely seems as though there may be issues with something like the oneiric numbers however.
In summary, what base theory would allow us to go further in the surreal construction? Additionally, how would we go about our construction & what types of objects would we run into beyond numbers, games & gaps?