Fleshed out to an answer since it got too big for a comment:
- 'things which are not of that form are called "pseudo-numbers" and usually ignored.'
- 'Is there research in which the pseudo-numbers are studied?'
These two statements seem rather at odds with each other, but yes — as Ted's answer notes, the subject you're looking for is Combinatorial Game Theory and it devotes quite a bit of study to these objects. While the order on games is only a partial order, there are games that can be compared with all numbers and do fall into the gaps you're speaking about - for instance, the game $\uparrow = \{0|\{0|0\}\}$ can be shown to be larger than zero but smaller than any positive number, even infinitesimal ones (this is a nice exercise!).
Note that even after plugging in games that can be ordered with respect to all numbers, there are still gaps that can be filled; this is usually done by taking $X_L$ and $X_R$ as proper classes rather than sets, although another tricky appraoch that shows up in Winning Ways is to give up the axiom of foundation (in their notation, go beyond the class of enders) and consider, e.g., games like $\mathbf{On}=\{\mathbf{On}|\}$ (which can easily be seen to be larger than any 'proper' game in a suitable sense). I believe this is equivalent for all practical purposes to defining $\mathbf{On}=\{V|\}$, where $V$ denotes 'the universe', but abandoning foundation allows games to be considered that couldn't be expressed otherwise.
Unfortunately, I don't know anything about the topological properties of these objects - I wasn't even aware that the Surreals had gotten much study in a topological sense! That's a subject that I'd like to know quite a bit more about myself.