The discussion in the comments kind of went off the rails, but the point I meant to make by linking to dichotomy between nice objects and nice categories is that you can get lots of examples by starting with a nice category and restricting to a full subcategory by imposing some condition on the objects that isn't preserved by categorical operations. Fields are one example; manifolds are another. So are CW-complexes and Kan complexes, or more generally the category of cofibrant and/or fibrant objects in any model category.

There's a certain genericity to this class of examples, of course, since any (small) category can be embedded in its presheaf category, which is almost maximally nice.