You say: where the objects are not some construction on sets or classes.  What do you actually mean here? I will give several examples of my interpretation of this, in some you may think it violates that criterion.

* Let $G$ be any group and form a category $G[1]$ with exactly one object which will be denoted $*$.  The set of morphisms from * to * will be $G$ and the composition will be the multiplication in $G$ with the identity morphism on * being the identity element of $G$.  That gives a category (in fact a groupoid). (You really only need a monoid not a group of course.)

* If $X$ is a topological space, the fundamental groupoid of $X$ is a category in which the objects are points in $X$ (does this violate the conditions that you imposed?), and the morphisms/arrows are homotopy classes of paths between the points.

* Let $(P,\leq)$ be a partially ordered set, and think of it as a category, i.e. objects are points, arrows are pairs $(x,y)$ with $x\leq y$. Similarly any equivalence relation on a set gives a category, in fact a groupoid, but again your question is not precise enough on what you want so you may feel this is cheating.

(Just in case the question resulted from a problem set in a category theory course, I have left you to write down ALL the details! In any case, it is a good idea to do so.)