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?