As background let me start by stating what I perceive to be the point of problem books, or to put the matter in perhaps more acceptable way, how I define problem books. A large majority of textbooks include exercises. Problem books have two distinctive features: the first is that they include at least hints and often complete solutions, ideally both; the second is that the problems are not routine exercises.
Exercises are primarily to make sure you have correctly grasped the notation, key results, key concepts etc. They are an essential test of understanding for the majority of students. On the other hand, a teacher or researcher in the field can typically solve exercises on sight. Certainly they can see how to solve them, even if some of the details might take a few minutes to work out. Most Questions on this site are exercises. The major group of exceptions (leaving aside the hopelessly ill-drafted questions) are the contest problems.
Note that the problem/exercise distinction is independent of level. Questions in several complex variable theory can be exercises, and questions about Euclidean geometry can be hard contest problems.
Contest problems have become an important subset of "problems" (under my usage of the term). However, a large majority of contest problems are at the teenage pre-university level (all the "olympiads" and similar contests). Two easily accessible exceptions are the Putnam exams and (less easily) the Miklos Schweitzer competitions.
Over the last decade or so, it has become more fashionable to publish problem books as supplementary material for university teaching. The earliest and best were perhaps the Polya/Szego Problems and Theorems in Analysis (2 vols) (still in print), and Halmos' A Hilbert Space Problem Book. As an undergraduate at Cambridge University in the late 1960s the Knopp problem books were assigned for the first complex variable course.
There are still some really excellent new(ish) problem books. Some of my favourites are Problem Book in Relativity and Gravitation (Lightman et al) - not so new (I see I bought my copy in 1977), Cosmology and Astrophysics through Problems (Padmanabhan) (purchased 1996), and Arnold's Problems (acquired 2005, but material originally published 1999).
Guy and Croft's books (Unsolved problems in ...) are not really the same. They are essentially convenient annotated references. But Chung and Graham's Erdos on Graphs is different somehow. I like it, despite my lack of progress with it.
I also like the multi-authored Algebraic Geometry, A problem solving approach, despite the fact that many problems are trivial. I found it a fast way into a field I had totally neglected. Similarly, Halmos' unique style carries me along in his more recent Linear Algebra Problem book despite the fact that many problems can be solved on sight.
Finally, I hesitate to mention names, but some of the problem books on number theory seem to me collections of exercises rather than problems.
So my questions are: (1) can anyone recommend any favourites to keep me amused on the dark winter evenings that are fast approaching ...? (2) clearly the viewpoint above is unashamedly elitist. I am worried about the future. Too many people are being brought up (right through toy problems at the PhD level) in a way that is unlikely to help them do worthwhile research. Is that viewpoint tenable? Should the best students (at least) be encouraged to tackle problems at the Putnam level as undergraduates? At the Miklos Schweitzer level? (3) is it feasible to give as homework, problems at the contest level? Or are they simply "enrichment material"? (4) The previous questions seem to be addressed primarily to teachers and researchers. What do students think? Would they like more problems? Or less.
Finally I am aware of several previous questions in this area. The closest seems to be Good problem books at a relatively advanced level? . But it is not the same question. Nor did it attract much interest :(