What's a reasonable category that is not locally small?

The category of multi-spans spans (thanks to everyone below for correcting my terminology). The objects are sets, and a map from $A$ to $B$ is a set $X$ equipped with a map $X → A × B$. The composition of $X → A × B$ and $Y → B × C$ is $X ×_B Y → A × C$.

I am stealing notation from algebraic geometry here: $X ×_B Y$ is the limit of the diagram $X → B ← Y$.

Admittedly, I've never wanted to allow $X$ to be an arbitrary set. I usually want it to be something like a finite set, a finite simplicial complex or a scheme of finite type. But it is certainly natural to define the category without any restrictions.

If C is a locally small category and W is a class of morphisms, we could try to form a category C[W^-1] by "formally inverting" the morphisms in W. The resulting category has the same objects as C, and it's sort of clear what the morphisms should be: some kind of zigzags of morphisms, where the backwards morphisms are required to be in W, modulo some equivalence relation (so that the backwards morphisms actually are inverses to the morphisms of W).

The trouble is, there will generally be a proper class of zigzags between any two objects X and Y; for instance there might be a proper class of objects Z which each give at least one zigzag X → Z ← Y. After taking equivalence classes, it's very unclear whether the Hom classes of the resulting category C[W^-1] are actually sets. In general, they certainly don't have to be.

Now there are very non-trivial techniques for proving that C[W^-1] actually is a locally small category in many cases of interest, such as hTop (as mentioned in a comment). So this is just an illustration of what might have gone wrong.

A one-object category consists of a class of arrows equipped with an associative unital binary operation --- namely, composition. It's a 'possibly-large monoid', if you like. And it's locally small iff this class is small (i.e. a set).

So, to produce a reasonable non-locally-small category, it's enough to produce a reasonable non-small monoid. The monoid of cardinals under addition is one. The monoid of cardinals under multiplication is another.

Cardinals are just isomorphism classes of sets, and we can produce similar examples by taking isomorphism classes in other categories. For instance, we could take the monoid of isomorphism classes of groups, with direct product as multiplication, or the monoid of isomorphism classes of vector spaces over Z/23Z, with direct sum as multiplication, etc etc.