Taking an extreme, we cannot currently treat a macroscopic phenomenon like the mind quantum mechanically. This is not because QM does not contain the essential feature necessary for yielding mental states; this is because the brain consists of billions of elementary particles and that's simply too many for any person or computer to handle. This would have been true of the boron atom fifty years ago, which I can now model on my laptop without too much hassle. Maybe one day my great great great great great grandchildren will be able to observe mental states in a computational model of the brain. But probably not since I have no children.

Unless we have a technological innovation to support handling stupendously large numbers of degrees of freedom, it is insane to treat a large complex system using more exact, elementary theory. The cutoff limit is a technological characteristic only. It is not in any way evidence that any non-elementary field of study is not reducible to a more fundamental theory. There is nothing wrong with a purely theoretical description of the entire Universe in principle. There's something truly wrong with anyone who'd try it in practise.

The other obvious point to make is that just because one set of laws is seen to emerge from a more fundamental set of laws (such as chemical and classical mechanical laws from QM) it doesn't mean that, on day 1, proponents of the latter can explain everything that the former can. There's a lot of ab rectum arguments that start: "But science cannot explain..." which belie a more anti-scientific than anti-reductionist stance. It is not science's job to provide answers to all questions -- it is not a religion. It's often science's job to say "I don't know, let's find out" -- it is a method of enquiry and verification.

The argument that e.g. physical chemistry is incapable of describing mental states on the basis that, still in its infancy, it has failed to describe mental states is a failure to note the trajectory of scientific accomplishment. It is not a "proof" that mental states cannot be yielded from more elementary theory. Perhaps the most interesting part of moving to a more fundamental theory is seeing precisely how complex phenomena emerge, if technology allows. It is certainly not desirable to "eliminate our most interesting phenomena"; we are simply limited in what we can achieve at any given time. Again, it's a method, not a religion.

Last point: science is reductionism-friendly, not reductionism-driven. When someone discovers a more fundamental theory, we examine it. We keep our minds open to the possibility that the electron has internal structure, while proceeding on the assumption that it does not, at least until that assumption can be shown to fail. It is nature that has shown herself partial to reductionism, giving us one phenomena (electromagnetism) that looks to us like two (electricity and magnetism), then three (electricity, magnetism, and the weak nuclear force). We have had to change our theories to suit her. The idea that science is reductionist endeavour because of its human constituents rather than because of the nature it represents paints us as far more motivated than we really are. We get points for testability, not originality.