I would second MMOs. Games in which the majority of people solo, with challenge tuned for this playstyle, are much more enjoyable as a duo. You have someone with you upon whom you can depend, to heal or tank for you, etc. You'll get through stuff faster, depend on each other, level together and get to develop your characters to each complement the other. If I could enjoy every MMO this way, I'd enjoy them a lot more. I'd recommend Conan/Secret World from Funcom as F2P games with some substance to them. A tank/healer duo is particularly great as you're entirely self-sufficient and doing any dungeon-type content (4-5 players) is incredibly easy as you'll never have to wait to get a team together.
Saints Row 3 is a brilliant game and in co-op, it really shines. Going through that game with a friend is a fantastic experience, especially if one or both of you give in and embrace the zany nature of the game. It adds an unpredictable element to the mix as they do things spontaneously you didn't expect or plan, often to hilarious consequences.
Splinter Cell: Conviction has a fun co-op mode. You each play one of Kestrel and Archer, there is a story of some sort involving Russians, split into four campaigns each of about 3-4 maps. You'd think a stealth game wouldn't lend itself to multiplayer with a partner, but it works surprisingly well. The two of you systematically disable lights and guards, resuscitate and rescue each other, disappear into shadows when one of you inadvertently alerts the guards (it will happen) and make clever use of both your gadgets and mark/executes.
Other multiplayer games worth mentioning: Borderlands, Left4Dead2, Resident Evil 5.
Other multiplayer games I've not tried with a friend (but would like to): Hunted: Demon's Forge, First Templar, Dead Island, Fable III.