Pretty much the video games that you cherish most, solely because of its story-line/characters and the such.
Mine has to be the whole Kingdom Hearts set. That story was so interesting and creative... it's crazy how I nearly shredded a tear at the end of KH2. It's the depth of that story that's really got me sprung over it.
And finally, Final Fantasy XIII-2.
Even though the game is crappier than manure itself, the story was so saddening and just great. Suffice to say I am in love with that game because of this special feature, without this jaw-dropping story;I'm not even sure if I would consider this game to meet my standards.