I guess why I'm asking is that I expect a lot of number-like-things to show up a lot across the multiverse, and that some of these numbers are going to be mathematically more interesting than others, just how Graham's number might be discovered by alien mathematicians but neither we nor aliens care about the number 315167427357825136347, which is basically infinitely smaller. An infinite ensemble is infinite but we expect math to be very similar in large swaths of it at least.

What's the smallest un-interesting number? But isn't that a rather interesting number...

Graham's number may be interesting to us and aliens a lot like us, but so what? I doubt it's interesting over all or even most of, say, a Tegmark level IV multiverse.

So my intuitions are reasonably confident that your proposal would only work if we're limiting our search to some set of quantities that doesn't include the very skewing mathematically interesting ones (like 'infinity').

Limiting your search is, I think, exactly why our current abstractions are a bad base for a universal prior like what is being discussed in the Mugging.