It seems like most of the books, and I haven't read that many, but whether Hungerford, Lang (basically a classic), Herstein, I'm actually partial to Fraliegh, which I learned from under Stallings, will have hard exercises. Maybe baby Herstein, if I can call it that, because he has a harder book, Topics in Algebra, or some such, would have problems that are a little too easy.
I think Milne has a book, though it may just be on group theory. There are, probably, alot of more specialized books. If you are into combinatorial group theory, I heard Magnus is good. Finite groups, Burnside. Representations of finite groups, someone named Bob Steinberg. Different than Robert Steinberg, the excellent UCLA professor, who discovered the Chevalley groups. Speaking of whom, his notes, in the back of Lie Groups, Lie Algebras, and their Representations, by V.S. Varadarajan, are quite good, as is the book itself.
One title I like is Rings and their Modules. I forget the author, but I would like to check it out. Modules are an important topic in Algebra. Among other things, the structure theorem for finitely generated abelian groups is a special case of the structure theorem for finitely generated modules.