The category of
multi-spans spans (not sure if this is the standard name)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 x B. The composition of X --> A x B and Y --> B x C is X x_B Y --> A x C.
I am stealing notation from algebraic geometry here: X 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.