Unique stories that develop naturally during gameplay, and are not imposed by the theme or objective.
The more open-ended a game is, the more opportunities players have to explore the mechanics (and interact with each other) in unique and unexpected ways. Different people will develop very different playstyles, thus leading to a story landscape of infinite possibilities. It's almost like each time you play these games, you are writing a new epic movie featuring tough choices and sacrifice, trust, and retribution. My favorite games always feature an extended post-game discussion, and many of the very best games can even lead to special moments that dwell in our memories just as vividly as any great movie scene or classic sports play. Games featuring a linear track that players have to move around to reach a set goal are not very interesting to me; I personally prefer games where the path is more unclear; where every decision each player makes has immediate ripple effects that may not ever be fully understood until long after the game is over. In other words, I want my games to be more like real life.
Take for example, the following rules primer to Chaosmos: