World of Warcraft.

I used to play Neverwinter Nights with my ex.


though I'm not sure what you mean by "not too gamey" :\

Borderlands, Saints Row 3 or Torchlight 2.

That.

EDIT: One of you could play a tank or a dps and the other a healer. You would compliment each other quite well, and going through the world of Azeroth together might bring you closer, in a way.

EDIT2: Or any other MMO, but I've not played anything else that would be good enough for me to recommend.

the first borderlands is a good one, its not difficult and has co-op mode. you could also look at RIFT now its gone f2p

The recent Neverwinter is quite friendly to poor computers, and it is surprisingly good.

I know at least three couples that play Minecraft together. They claim it's perfect.

I'd recommend Neverwinter Nights as well. Terraria or minecraft can be good as well if you want to know some aspects of her personality better :p

Great new RPG called, "While You've Been Away I've Been Sleeping With Your Sister".
The graphics aren't that great but it's got a nice interface and you don't need a super new high end pc to run it maxed out.

Me and my partner had three "teams" we used to play, and which works surprisingly well: Priest-Hunter, Paladin-Mage and Hunter-Hunter. With any of these combinations we soloed almost all content except raids.

If the person is not a gamer or hard-core gamer then your looking at minor MMO, Sandbox games or Facebook stuff

Minecraft - yep
GW2 - possibly

I would possibly stay away from FPS as I struggle with those sometimes and a hard-core gamer will use you all as headshot fodder.

So look for sandbox type games where you build things together etc

Puzzle Pirates is relaxed and based around minigames. You can own and furnish a house together. You can buy her clothing items and wrap them in gift boxes. You can even buy her a ship and name it after her, if you're clever about it. You can get portraits of your pirates together, too.

I'm unsure how active it is now, but I expect there are enough people on for small-ship voyages. The time I spent there was pretty happy. The gamemasters (oceanmasters) are active and patient, and often host interesting competitions, both in-game and on the forums.

Get yourselves an FUFME kit... plenty of fun to be had there :P

solution for the Internet and corporate intranet.

hmmmmmm.......