My definition for game:
A game is a piece of art in which a player or players interact with the game itself or with each other under a set of artificial rules which allow the players to complete goals and to fail at it, depending on how the players interact with the game or each other.
By my definition, Cookie Clicker isn't a game because you can't lose. Tag is a game, War the card game (without a variant or house rule that adds interactivity) isn't, War the... war isn't, a tabletop RPG is, most slot machines are but not all of them, and I don't know what Farmville is so I can't tell if or not it's a game. And, quite ironically, The Game is not a game because you can't win (and by the way, you just lost it).