I think the careful choice of title is one of the many strokes of genius in this excellent short story. The word "game" definitely operates on multiple levels, referring ironically to a "game" that children might play, but indicating how tragic and dangerous this game is, referring also to the most dangerous form of "game" on the planet - man, and also indicating the life-threatening battle of wits and skill that Rainsford and Zaroff undergo. Fascinating use of irony.
How right everyone is about the multiple layers of meaning in this story. The most dangerous game is that which Zaroff is playing as he turns human beings into prey. And yet Rainsford becomes not just part of the game, but the most dangerous game Zaroff has ever hunted.
I have always thought that the meaning of "game" in the title goes beyond the double meaning to perhaps include a third. In the opening scene, Rainsford clearly states that he think hunting is the greatest sport (or game) in all the world. However, once he becomes the hunted, he quickly sees that this is not a "game" at all. That word just has such a light connotation; it fails to really capture the violence and terror in the hunt. It is not until Rainsford becomes the hunted that he sees this and realizes that the "game" is really a "fight" for survival.
An excellent question! "Game" refers both to the macabre game General Zaroff plays when he hunts humans and to the human prey that he regards as game just as he regards the animals he has killed as game. The "most dangerous game" then is one in which humans are hunted for at least two reasons: human life is at stake, and human game is dangerous because humans are the only species that can reason; therefore, humans pose a greater challenge for the hunter. In Zaroff's case, his game with Rainsford proves to be deadly for the hunter this time. The hunted outwits the hunter and manages not only to survive but to kill his pursuer. The game is over for Zaroff.
This was actually one of the most interesting pieces of irony for me in reading the story. At first glance, the word game seems to refer to the idea of the intellectual competition between Rainsford and Zaroff, but by the end of the story you realize that Rainsford himself is the most dangerous "game" meaning the most dangerous hunted animal, as he hunts down and kills the hunter.