Does natural selection always maximize mean fitness?

A question of interest in theoretical population genetics is whether natural selection always drives the population to the state at which the mean fitness is the maximum possible. There are cases in which it does not:

Frequency-dependent selection is an example. It can maintain a polymorphism, and when it does the fitness of a genotype is highest when it is rare. But when a genotype is rare, natural selection acts to increase its frequency, making it less rare. It is possible to devise theoretical cases in which the mean fitness is maximized at genotype frequencies that are not at equilibrium: natural selection acts to reduce mean fitness by moving the genotype frequencies away from those points.

There are many cases in which natural selection maximizes simple mean fitness, and for many purposes this is a useful assumption. We can then think of natural selection as a hill-climbing process, by analogy with the hills in the adaptive topography.