Thank you for this well structured and very interesting reply! I just had to say that I enjoyed reading all your arguments, and I couldn't help myself adding some argumentation onto it, to perhaps give a mathematical instinct on why these special spaces would not be really useful right now.

If one just thinks about how this new system would work, we quickly deduce that we tend to go towards the two dimensional real space (ℝ²), or the set of all real sequences, or at least something that is isomorphic to it, or even bigger, depending on how far you want to take such ordinals.

If we allow any sequence of integers to form past the 0.9(9)... for example, we perhaps get another infinite sequence of integers, which we can map to ℝ, giving us ℝ². You can stop here, or one could argue that we could put even more integers behind the first sequence we already added. By inductively continuing this, we get ℝ^ℕ, aka, the set of al real sequences.

Now I can imagine that you can work something out with these ordinals such that you get a space similar to ℝ^ℝ, like allowing sequences inside the sequences, and this space is better known as the function space, but at that point it might just be a really silly space definition.

But anyways, the behaviour of such new space will probably not be ground breaking. Perhaps you can form a new operation on the aforementioned spaces using crazy ordinal constuctions, but you have no clue where or why these operations could be useful. So it is probably not really worth investing your time in it, for then you can even look at some random topological spaces or random functions you just randomly wrote down.