This is a list games I own but have not finished playing/started playing. This list will probably be longer than I'd like it to be, but I think it will help to remind me to finish things.
Final Fantasy III (DS)
Final Fantasy: Dissidia (PSP)
Jeanne D'Arc (PSP)
Legend of Zelda: Twilight Princess (Wii)--I recently realized that I never actually beat the final boss in the game, but instead continued playing around in the world collecting goodies since I knew that after the boss I would go to credits or what not.
LocoRoco (PSP)
Mario Galaxy (Wii)
Okamiden (DS)
Red Dead Redemption (PS3)
Rouge Galaxy (PS2)
Valkaryia Chronicals II (PSP)
Wizado (DS)
What "finished" means to me
Personally I am officially "done" with a game when I beat the main story mission. Some games are put down and never picked up again after I "finish" them, but other I choose to continue playing because I enjoy them, and wish to further progress the extras in the game.