Note: I'm literally just going through my achievement list and pulling exclusives out so you're going to get an...interesting mix here
Catherine: Granted, I just started this game so maybe it turns into a turd but this is the quirkiest of quirky games I think I've played. Basically you're a guy who has a girlfriend named Katherine. Another girl named Catherine seduces you and now you have yourself a pickle! Half interactive story (like those Japanese Story Games...they have a name but I can't for the life of me think of the genre name) and half puzzle platform (you have nightmares where you have to climb a pyramid to escape some kind of monster). There is a demo of it available but so far, I'm enjoying it but again, I've only played about a hour.
Resonance of Fate: Keep the weird train going with this RPG. It's a traditional JRPG-story but with an incredibly weird and somewhat complicated battle system. Honestly, it took me quite a while to really get a hand of the fighting system but if you can, it's a really cool game. The story is...a'ight. You're namely playing for the interesting characters and the interesting fight system.
Crackdown: Grand Theft Auto with super powers. The story is virtually non-existent but it's still a pretty fun game. It launched with the 360 but it's still a pretty fun game, if a little dated. Crackdown 2 was a'ight but it was not as good as the first game.
Operation Darkness: Good luck finding it but Atlus makes another appearance (Catherine). A tactical RPG which combines WWII and the occult. Basically monsters exist. You're a werewolf and you are in the army to fight Hitler's magical monsters. A questionable camera hurts this game but I thought it was a really cool twist on the TRPG genre.
Lost Odyssey: You mentioned it already but I'm mentioning it again. If you like JRPGs, you should play it.
Stuntman Ignition: Holy balls we're getting back in time here. Have you ever wanted to be a stunt driver without having to put yourself in harms way? Then have I got a game for you! It's a blast to play and can be surprisingly fun/frustrating as you try to get the perfect run through the movie scenes.
Condemned 1 and Condemned Bloodshot: If you ever need proof that FPS fistacuffs will work, look no further. One of the most genuinely creepy games I've played. You play as a cop in the middle of a bunch of people going crazy and wanting to bash your skull in with a bunch of random stuff. While Bloodshot went into crazy-land for the ending, both games are still worth playing if you're into scares and pissing yourself.
Hope this helps. All of these games are older so you should be able to get them pretty cheap. I stuck exclusively with the "not obvious" ones so most of these are either pretty weird or rough around the edges (or both).
