Minecraft far and away is the game we play together most of the time. Alas it has finally gotten old to us.
One Christmas we played eve online. That was a great bit of fun. Bit of a time sink though.
I wish The Long Dark had coop.
Warhammer Vermintide 2, Left For Dead 1/2 or Borderlands 2 for FPS coop. Dead Space 3 if you don't mind a few scares. Ark is decent for PvE coop.
Portal 2 for puzzles. Rocket League for sporty action (albeit coop vs other people). Throw in Factorio for building.
Path of Exile/Diablo 3/Torchlight 2 for ARPG. StarCraft 2 for RTS.
But overall I'm with you, I feel all games can be enhanced with coop.
I don't think there is anything wrong with competition. I think it's the company that makes the experience.