One common definition of a category features objects, arrows, an associative composition law for arrows, and an identity map requirement: for every object there must be an self-pointing arrow that behaves as left-side and right-side identity for composition.
I have no problem seeing how one could use a formalism of objects and arrows between them to structure our thoughts on a wide variety of areas. Also, it's easy to see why one would want to have a basic building operation (i.e. the composition law) in such a formalism.
With a little more thought I can also see why one would stipulate that this composition law be associative: making composition associative has the effect of replacing the "pair of arrows" with the "sequence of arrows" as the formalism's basic building operation, and I can see that this change may make the formalism more flexible.
I can't come up with any rationale at all for requiring that there be an identity arrow for every object.
What is lost by dropping this requirement?
Note that dropping this requirement does not preclude the existence of identity arrows, and therefore, it does not preclude defining concepts (e.g. inverses, isomorphisms) that depend on the concept of an identity arrow.
In fact, AFAICT, if one were to drop the identity arrow requirement, one could still define functors, natural transformations, cones, etc. (except that the definitions, of course, would not have the usual stipulations for the identity).
EDIT: To sharpen the question a bit, and maybe make it less of a "philosophical issue", I note that several times I've come across remarks from Saunders Mac Lane to the effect that their aim (his and Eilenberg's) in coming up with category theory was to arrive at the concept of natural transformation. In fact, in Categories for the Working Mathematician, 2nd ed., Mac Lane writes (p. 18):
"category" has been defined in order to define "functor" and "functor" has been defined in order to define "natural transformation".
As I noted earlier, the definitions of functor and natural transformation don't seem to rely much on the identity law requirement for categories.
So I could make my question a bit more concrete: how does requiring that every object in a category have an identity contribute to what Eilenberg and Mac Lane were aiming for?