What are some intuitive examples of categories which aren't made from structured sets and functions between them? In my loose reading about category theory, categories are usually introduced in terms of sets, often with some structure (such as algebraic or topological), and functions between them that preserve that structure. I find this easy to intuit, and makes ideas like commutative diagrams easy to understand.
However, I know that categories can be much broader than this, even before we get into higher notions like functors and such, and that categories describing structured sets and the functions between them are only an important but small portion of them. Despite this, I find it hard not to unconsciously interpret diagrams and category theoretic statements in these terms unless I completely strip all meaning from the symbols and treat the whole thing as a purely syntactic game, and the times where I've seen people discussing categories where the morphisms aren't structure preserving functions have been very abstract and gone right over my head.
So what I would like are some simple, intuitive examples of things which can be described by category theory, but where -
A) The objects are not sets,
B) the morphisms aren't (possibly structure preserving) functions, or
C) both.
I've wanted to learn more about category theory for a while, but I've found I kept being held back by my lack of a strong, accurate mental model for the sorts of things categories can represent, in the way that I have a strong model for the ways sets work and the things they can represent.
 A: One of the standard introductory examples of a category is "any given group", which can be implemented as a single object, an arrow for every group element, and arrow composition defined by group multiplication. I think of the object as being a blob of state, and the arrows as ways to act on that blob of state.

This generalises vastly. One of the viewpoints which is sometimes helpful (for instance, in formulating and proving the Yoneda lemma) is that a category is nothing more nor less than a template of a many-sorted algebraic theory with unary functions. Each object represents a type, and each arrow represents a function from some type to some other type. (The types aren't "implemented": the category doesn't know that the object which you've labelled $\mathbb{N}$ actually contains anything, let alone the natural numbers, since it's only a template.) An instantiation of this template is a functor $\mathcal{C} \to \mathrm{Set}$. I've written an incomplete blurb about this, inspired by https://mathoverflow.net/a/15143.
A: Preorder $(X,\leq)$ is an example. 
(Here $\leq$ is a reflexive and transitive relation on $X$)
Objects are the elements of $X$ and a homset contains at most one arrow which is not recognized as a function. 
Such arrow exists in $\mathsf{hom}(a,b)$ iff $a\leq b$.
A: There is a common example that is a bit of a cheat (can you see why ?) which consists in having natural numbers as objects, and arrows from $n\to m$ be $m\times n$ matrices (say real coefficients), and composition is just multiplication. 
An example that is less of a cheat and that is actually super useful is the fundamental groupoid of a space $X$ : $\Pi_1(X)$. This category has as objects points of $X$ and as morphisms $x\to y$ homotopy classes of paths that start at $x$ and end at $y$. 
This is a category-theoretic version of the fundamental group (in fact for path-connected spaces, it is equivalent as a category to the fundamental group) but it's more suitable for various purposes. 
A: There are a lot of examples, of such things
If you are familiar with type theory, you can build a category whose objects are contexts and morphisms are substitutions between them. 
If you take a set with a reflexive and transitive relation, you can construct a category whose objects are members of the set, and there is exactly one morphisms between any two objects that are in relation
In general, the categories that are in the flavour that you mention are called concrete category, and an important result is that the category whose objects are pointed topological spaces up to weak equivalence, and morphisms are induced by continuous maps (to be precise this is constructed by taking a construction called as localization) is not a concrete category
Interestingly, if you take a directed graph, you can construct the "category of paths" (not a standard name), whose objects are the vertices of your graph, and morphisms from $a$ to $b$ is the set of all paths from $a$ to $b$ in the graph. It is usually called the category freely generated by the graph, and is not constructed as a concrete category (although I cannot exclude that il some cases it might be equivalent to a concrete category)
