There are many many definitions of a game, most of which differ only subtly. Some are dependent on the definition of play, but that's a whole other discussion. Maybe I should have written about that first, oh well. Anyways, here are a few ways to define a game, accompanied by thoughts:
Katie Salen and Eric Zimmerman define a game as a "system in which players engage in an artificial conflict, defined by rules, that results in a quantifiable outcome".
Conflict, rules, and a goal are the driving factors in about all definitions I've read. I think the unique part of this one is "quantifiable outcome" which would mean a goal. But saying "outcome" instead of goal, seems like you could get there without necessarily meaning to. I guess technically, there are games (like the card game War) you could win by just following the rules with no ambition.