What are some examples of naturally occurring badly behaved (possibly higher) categories?
When working with a specific category like ${\bf Set}$ or ${\bf Cat}$, we usually understand/explain them by lauding structural properties they posess — ${\bf Set}$ is an autological topos, ${\bf Cat}$ is Cartesian closed, etc.
What structures can be arranged naturally/canonically into (possibly higher) categories, despite the resulting categories having few/none of the structural properties we would usually like to have in order to carry out category-theoretic-type proofs?
A natural example is ${\bf Field}$, the category of fields and field homomorphisms, since it has no terminal object, no initial object, no finite products, is not algebraic and is not presentable. (It is, however, accessible with a multi-initial object given by the set of prime fields).
I suspect that it gets worse than this, but I can't think of anything further off the top of my head. Any examples are appreciated.
Best Answer
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.