Normally I'm fairly good about finishing what I start (ignoring stalled replays of games, of course) if I've played any of it all (read: ignore games dropped in the first hour), but I've gotten worse lately. At least that usually means "buy games and don't play them" now, I know not to try and play 5 games at once.
Have played in 2013 but still haven't finished:
X-COM - Enemy Unknown: I had fun with the game! Just screw waiting for the 2nd plot mission with the Oracle ship to show up in the right continent. I'm maxed out on Colonels, just let me freaking advance the plot without doing stupid missions that don't get me anything useful now.
The Last Story - I'll definitely finish this though, but taking a break from it for awhile.
FF13-2 - I don't know if I'll finish this! Only halfway throughish, though.
Potential infinite backburner:
Mega Man 10 - I enjoyed this game too, but got into Wily stages, started writing up detected weakness of the MM1-9 cameo robot masters, realized I shouldn't use consumables, died, realized I should theoreitcaly grind a bunch for screws to take on Wily, thought that was lame, never went back to it.
Avalon Code - I don't think I got fair enough to call this one "nearly beat", but I got ~20% of the way through this or so (wild guess?) which is usually enough to hook me, but dropped it. Unlikely to return.
Finally, the big one, been on the backburner for ages yet might still be returned to some day:
Tales of the Abyss - Hey, I love this game, I have it as 9/10 in the RPG Ratings topic. I just don't need to actually, uh, finish it. I like Tales gameplay but can only take so much. (I know that for Symphonia I was pretty much running away from every single enemy in the final dungeon to finish the damn plot. Despite liking Abyss's plot more, I still wanted it to just hurry up and end.)