Next is Pontryagin's: the category of locally compact groups. The original Pontryagin duality easily generalizes: the category of locally compact modules over a given commutative ring is self-dual. (In the non-commutative case one also obtains a duality but not a self-duality -- unless, of course, the ring is self-dual.) A corollary is that the category of discrete left R-modules is dual to the category of compact right R-modules. (For 50 years I've been trying to turn this into an exercise in abelian categories. There's a nice reduction down to the proposition that R/Z is a cogenerator for the category of compact abelian groups, but that fact seems to require some non-trivial functional analysis.)
This reminded me of a long-standing torture: does anybody know an elementary proof at least of the particular case when the base ring - thus the dualizer too - has only two elements? (Demanded by Guram Bezhanishvili; I agreed to try thinking on this one as it seemed somehow close to Boolean algebras, but...)
Then, of course there's my present favorite: the category of finitely presented group-valued functors from the category of finitely presented modules over a commutative ring.
Yes, yes?