A lot of games require two hands, one on the keyboard and one on the mouse. First-person shooters are the prime example of this, where the left hand is on WASD and the right hand aims and shoots. Real-time strategy games such as Starcraft require keyboard use for all but beginner play. Likewise with World of Warcraft and its many clones. Even my previous Flash MMO project Realm of the Mad God needs both keyboard and mouse.
I'm interested in exploring the idea of "sandwich compatibility." A sandwich-compatible game can be played with one hand, while eating lunch. One could argue that the sandwich compatible game was invented by the Earl of Sandwich. Actually, he invented the sandwich itself, but the point is that he wanted to play games and eat at the same time.
There are plenty of Flash games that need only a mouse, and that's one reason these casual games work so well -- you can play them on your lunch or coffee break. Tower defense, match-3, physics puzzlers -- none of them require a keyboard. Munch....click....drag...munch...click...
Related to the sandwich compatible game is the "bus stop compatible" game -- something you can play with one thumb on a smartphone while standing at a bus stop and holding a bag of groceries in your other hand. Angry Birds is a great example.
I'm working on a sandwich compatible MMO. I want to step back from the complexity of games like WoW and the hectic frenzy of Realm of the Mad God, into a more relaxing place. Dialing down the pandemonium and leaving one hand free lets people play more often, for longer at a time. And maybe someday the game can find a spot at the bus stop too.