For me it's:

1)King of Fighters:

Excluding 98/02, this game has had a huge story tie in to all its games. With the casts of the game topping out at really high numbers for the majority of the series, each character is given extensive story consideration. Just look at the Drama cds. Each overall story conflict, individual character development, and overall team story is considered with each game. Not to mention the sheer amount of character development that goes on with character that are present even outside of the series (Terry, Robert, etc). The characters are also very distinct in their personalities even in gameplay. With their individual voice acting, character interaction, and even walk animations (Vice, Yamazaki, etc).

2)Mortal Kombat:

Mortal Kombat has taken pride in its overall story development. Each character gets their own backstory, and many of them have developed immensly since their first debut. Characters have truly defined themselves in the series. Mortal Kombat still puts a huge impact on the overall story of the game, which seems to effect the overall feel of each game.

3)Guilty Gear:

Although still relatively green...Guilty Gear has developed a huge backstory and conflict with its characters. The majority of the story is well thought out...although most gamers miss a glimse of it since it exists officially more outside the game then anything.