Is a mixed strategy ever the best response to a pure strategy? Suppose you are playing a game against an opponent whom you know only uses pure strategies.  My question is, is there any such game in which using a mixed strategy in response is better than all the pure strategies you have at your disposal?
Fixing the strategy of the opponent, a mixed strategy never yields a strictly higher utility if you are expected utility-maximizing.
The reason is that the expected utility from a mixed strategy is at most as high as the highest utility from the pure strategies which this mixed strategy plays with positive probability.
That is not to say that a mixed strategy cannot be a best response, though. 
Consider for example rock-paper-scissors and a imagine your opponent playing each of the three actions with equal probability.
This leaves you indifferent between all three actions and hence any mixed strategy is a best response.

Things can be different when players are not expected-utility maximizers. If that's what you're interested in, you could take a look at the paper "Equilibrium without independence" by V. Crawford, 1990, as well as some of the references therein.