Thanks, I think that's a really good answer, and sort of in the spirit of what I was thinking about. I had not heard of the term "designer homotopy theory." However, I should hope that, after swinging to the side of purely algebraic knowledge, in time, we could come up with nice ideas of what such things look like. However, after writing this question it occurred to me, knowing what such a thing "looks like" may in some sense be the same as knowing AT LEAST its homotopy groups, which we don't always know!

This actually happened to me recently. I told a topologist about Hovery and Palmieri's Dichotomy Conjecture and he said "That's not known??".

@Dylan Thanks! This is a great answer. I think that I should have been more clear in that the things I am really interested in are localizing subcategories and Bousfield lattices, which indeed, is stuff that Luke knows a lot about. Sometimes I forget that there is a lot more relevant information to categories besides their localizing subcategories ;-). I have read a little of Luke's work, where he shows that the situation can get really bad. I guess I just wanted to see if there was something general like HPS section 6 for non-Noetherian rings, but I think not.

@David that sounds like exactly the type of thing I would like to know about. All in all, it sounds like the answer to my question is: no. There does not seem to be much of a general theory.

@Fernando thanks! I've read Neeman's paper, and a bit by another fellow Luke Wolcott. This is a really interesting topic! Seems like for non-Noetherian rings stuff gets complicated quick :)

I have also just discovered an extremely brief discussion of it at: math.uwo.ca/~schebolu/apping/UW/future-research.pdf

@Fernando Muro No in fact I don't! I'll look into it. I guess I was just wondering if there was some specific algebraic obstruction, i.e. is there a stable homotopy structure on the derived category of a non-Noetherian ring?

Thanks Tyler, that's a really nice counterexample. This may not be such a bad thing. That is, I'd really be interested in knowing more about this limit of categories. For instance, what do its localizing subcategories look like? We know the structure of localizing subcategories for each category in the sequence. Also, can we get a more general "chromatic convergence" type situation by just looking at a limit of the E(n) local subcats, and not even their finite objects.

Of course assume we've already localized at a prime $p$.

I suspect it's something like the fact that all $E_1$-acylics that map to $X_1$ also map to something which is equivalent to $X^{1,2}$, and so at least their acyclizations are equivalent. Still working on it though.