A minor correction:
A small category C has finite products just when the representables in [C^op, Set] are closed under finite products. It is cartesian closed just when the representables are closed under finite products and internal hom.
It seems natural, therefore, to consider a notion of "cartesian closedness without finite products": categories in which the representables are closed in [C^op, Set] under internal hom but not necessarily under finite products. This amounts to giving, for each pair of objects X and Y, an object [X, Y] together with a universal natural transformation C(-, [X, Y]) x C(-, X) -> C(-, Y). Such categories will be closed in the sense of Eilenberg-Kelly without necessarily being monoidal:
On reflection, this bit isn't necessarily true, since EK-closed requires the unit object to be representable as well. Richard