Amagami

Amagami is a romantic simulation game, that features a unique system, that allows you to progress with the story in a non-linear fashion by choosing various tiles on the 'scenario' map in desired order. After choosing a particular tile you're taken to the corresponding scene and successfully completing it usually unlocks more tiles allowing you to further progress with the story. There are six main characters in the game, and one extra character that's unlocked after meeting certain requirements. The time limit is forty days, in that time you have to become close enough with one of the characters, otherwise you reach a bad end.