@JohnWiltshire-Gordon Yeah I wish. Haha. That's sort of my starting point. In a sense.

@S.Carnahan only to homotopy theorists. to everyone else it's still just a wedge

I should mention, I have been informed that there is a theorem somewhere (and I'm pretty sure I read it at some point) that the only finite length formal group laws are the additive and multiplicative ones.

I mean, I guess it's more of a "program" than a problem. The idea being that if we let $\hbar$ go to zero we get the "classical limit." But if we take some "infinitesimal neighborhood" of letting $\hbar$ go to zero (i.e. a deformation) we get some that sort of is like quantum mechanics, or something. I don't really know any physics, lol.

Deformation quantization in physics is a pretty amazing picture. Dunno if that fits your criteria.

I can't remember the details off the top of my head, but I'm somewhat certain this is covered in Lurie's Higher Topos Theory (arxiv.org/pdf/math/0608040v4.pdf) I believe on page 295. Might just be a good place to start.

I think there are a lot of good examples of such spaces coming from lattice theory, as the Stone spaces associated to Boolean algebras.

Aha, yes. Okay. I was overcomplicating things I think. The $X(n)$-module structure on $X(n+1)$ is not complicated, and one can take advantage of the Thom isomorphism on $X(n+1)\wedge X(n+1)$ as well. I'll try to make sure what I've written down makes sense tomorrow. If so, I'll post it, though, of course, I'd love to see what others have to say.

Yeah, pretty much. I'm currently looking at some oldish papers of Mark Mahowald on Thom spectra which are ring spectra, and specifically Thom spectra coming from loop spaces (hence $A_\infty$-ring spectra) to see how he does this kind of thing. I think I'm close to understanding it (though I've thought that about plenty of things before, and been miserably wrong).

Also, do you happen to know of good code (I guess in Sage, which I have little experience with) for computing the homogeneous degree n terms of the associator? I'd like to show that the associator is a cocycle in a certain cohomology, but am having a difficult time writing down these rather large polynomials for anything higher than homogeneous degree 2.