= Category of ordinals in reverse order
Regard the proper class of all <ordinal>[ordinals] as a poset category with $\alpha\to\beta$ when $\alpha\geq\beta$ in the usual order. It is locally small and complete: the product of any set-indexed family is its ordinary supremum, and equalizers are automatic in a poset. It has no initial object because there is no largest ordinal. Consequently every representable functor from it to sets preserves all small limits but has no left adjoint, since a left adjoint would have to send the empty set to an initial object.
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