L4D and L4D2 are solid games which are quite fun but the lack of official maps makes it quickly repetative.
Killing floor and Killing floor 2 are both fairly good 6 player coop games vs AI, the original looks kinda dated now and I'm not sure if the new one is out of beta yet, but they're both a good laugh and have a lot of maps and strategy, a bit grindy at first though as you have to level up your classes to make them good enough for the harder settings.
Payday and Payday 2 are quite good but people should think carefully about buying their games after the controversy they caused a while back regarding DLC, I personally wouldn't support this studio anymore.
If you're OK with older looking games there's a huge amount of coop value you can get from Sven Coop for Half Life, while the game is old they have an insane amount of maps created by the community that range from everything from puzzles to battles and small story campaigns, Sven coop is a free mod and HL is cheap.
Zombie army trilogy which is a sort of expansion for sniper elite is a long 3 part campaign where you play coop, it's 3rd person action with 1st person sniping which we had a lot of fun playing at a LAN.
Blackwake is a kickstarted multiplayer team vs team pirate ship battle game but is quite cooperative because you have to load cannons and do various tasks to make the ship combat effective and that's quite good with a bunch of you all running about. Be warned it's still alpha though and currently only available during scheduled alpha testing weekends. Lots of fun though.
While mostly singleplayer I thought that the missions in The Division where were most of the fun was, 4 player coop in lots of different scenarios in a kind of 3rd person/1st person aiming shooter, plays a bit like an MMO lite. The missions/dungeons are good fun with several competent players.
I'll give a shout out to several asymmetric multiplayer, Damned which was a kickstarted thing and is 1 monster vs a load of survivors, that's quite exhilarating and terrifying played coop however last I checked not many people were playing.
Like above the mod Hidden for HL2 is an asymmetric multiplayer with 1 person playing an invisible monster which the others hunt cooperatively that's always been a staple of our LAN parties.
Sanctum and Sanctum 2 are 4 player 1st person tower defence games where you build your base out of large cubes in between rounds and then help defend with guns during the waves and they're quite good as a mix between strategy and 1st person skill.
I've played Rainbow 6 Siege recently with friends and there's coop modes against AI terrorists for that which are OK but honestly the multiplayer is better although can be quite harsh until you've got used to it. The old R6 Vegas games were good as coop campaigns as well although they're 3rd person.
That's most in my steam list although quite frankly over the years i've fond coop mods to be better for straight out coop campaigns, there are others for HL2 that escape me right now, if you google you'll find a bunch.