Action/shooter coop: Shadow warrior 2(fun nonsense hack n slash + stereotypes) Alien swarm(keep eachother safe and work together, top down shooter, can be abit hard but its fast, its fun and abit frustrating at times) killing floor(zombie shooter on lsd whenever theres a season of smth(christmas, new years etc)) livelock(abit like alien swarm but its robots and its a more futuristic world with future tech weapons), castle crashers(sidescrolling hack n slash adventure) dark souls series(technically semi-coop, action rpg where you can summon your friends to help you in an area where you havent defeated the boss yet) pubg(fps/tps solo, duo or in a squad, the coop experience is very different from solo, and can be really entertaining and backstabbing) vermintide(left4dead but with warhammer, and all the zombies are skaven(Ratmen on steroids) and the apocalypse of rats overflooding the world is coming, good luck) space hulk deathwing(warhammer 40k left4dead ish, you are less of a survivalist more of an elite team sent to recover old forgotten tech from a giant ruin of a space ship/multiple ships that has been infested with tyranids, you sound heavy and so does the weapons, satisfying to play) sanctum 1/2(action fps + tower defense),
rts coop campaigns:
Battleforge ofc Dawn of war II(rts + rpg ish coop campaign) starcraft II(kinda late but the coop mode is okay, but limited to single instances(almost like a little coop custom map)) total war warhammer I/II(coop grand campaign in a civilization(X4)+rts hybrid styled map where you control your armies, provinces and diplomacy on a grand map, and then when your armies fight it turns into an rts and the graphics in this game is really just great.)
dungeon crawler? coop: Victor vran (fight against all sorts of monsters, loot simulator, jump like a ninja off walls to get up to higher levels - fun) diablo series (3 games all pretty good, first one is abit dated and you can feel its old, 2 is a masterpiece of a loot simulator and 3 did a really good followup(after a few big changes and some fixes) but overall a great overachieving story and fun to hack n slash your way through hordes of enemies with your friends) torchlight II (kinda like diablo 2 but abit more cartoony graphics and Fishing, with bombs)
Puzzle coop:
trine 2 (really fun puzzle coop where you control different characters with different abilites to help eachother get through the levels, sidescrolling) forced (top down ish puzzle coop where you and your friends fight together and try to solve the puzzles by working together)
turn based coop: Civilization (pretty fun to play together with friends against friends or against ai, time consuming) Endless legend/space, Eador, Warlock(all of these games campaigns are grand strategy mixed with smth like heroes of might and magic, really time consuming but really entertaining to play, you get heroes with armies and heroes and units can individually level up, in the endless series you can create your own units(more or less theres templates to it) Heroes of might and magic (not really a coop campaign but many fun hours can be had in multiplayer working together against other players or the AI)
Rpg coop: Nox (fun dungeon crawling ISH coop, but put it here instead because its not that much a loot simulator, theres good puzzles, good fights, fun bosses, you can murderhobo a decent amount of normal npcs and your friends if you feel like it, the story is entertaining and its a game just made for fun by fun)
Baldurs gate I/II, neverwinter nights I/II (oldschool read a lot Dungeons and dragons inspired rpgs, great stories, needs patience to read, especially with multiple people, takes a lot of time to play)
divinity original sin I/II (take the above games, add a good amount of slapstick comedy and A lot of enviromental spells(FUN) and you get divinity original sin enhanced edition, improve a bit on that and you get divinity original sin 2, in the first you can only play 2 players coop(4 with mods) in the 2 you can play up to 4 people coop, you can also betray your friends in 2) Elder scrolls skyrim(technically with mods :D) Chill/Funhouse/crafting/Mods coop:
Minecraft
Terria
stardew valley the forest wc3 custom maps, dota 2 custom maps, starcraft 2 custom maps(those 3 games are gold mines of short or longer coop-maps)