Final Fantasy 8, 9 & 10 for me. Don't play them as much anymore, still enjoy playing them though, but as a kid I'd replay these on a regular basis.
Pokemon. I just love Pokemon. And no interest in the competitive side whatsoever so that doesn't leave much else other than replaying the story, don't mind that though since it's pretty enjoyable.
Dynasty Warriors. I'm a huge musou fan, so I'm always up for replaying some DW!
Me neither, seems like every person I know (In person, not the internets, lol) other than my girlfriend and myself, loves to play Pokémon competitive, I remember my brother restarting X/Y over and over because he "needed" a Froakie with a specfic nature, lol. I don't really care about that stuff, I pick a Pokémon, I catch the ones I like, maybe a few others to help me get by some gyms, and I'm good... I play for the story, which is always great despite it pretty much being the same formula since the beginning, lol.