What's a reasonable category that is not locally small? Recall that a category C is small if the class of its morphisms is a set; otherwise, it is large.  One of many examples of a large category is Set, for Russell's paradox reasons.  A category C is locally small if the class of morphisms between any two of its objects is a set.  Of course, a small category is necessarily locally small.  The converse is not true, as Set is a counterexample.
Now, I can construct categories that are not locally small.  However, what's the most common or most reasonable such category?
 A: The category of Grothendieck topoi and (equivalence classes of) geometric morphisms. For example, if $A$ is the classifying topos for abelian groups, the geometric morphisms from $Set$ to $A$ are in bijection with isomorphism classes of abelian groups, which is certainly not a set.
A: An important class of examples where it's an issue for which a lot of technology is involved to get around is in the localization of a non-small (but usually locally small) category with weak equivalences C:
the morphisms in the localization are in general arbitrary finite sig-zags of morphisms in C (see simplicial localization and references given there). So if C is not small, this is a priori not locally small.
But in particular IF the category with weak equivlences hapens to extend to a model category does it follow that the localization, i.e. the homotopy category is locally small after all.
But if not, certainly one would always regard the localization as a "reasonable category". 
A: 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.
A: 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: 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.
A: The category Cat, whose objects are categories and whose morphisms between two categories consist of functors.
Whether this is "reasonable" is up to you to judge.
