Up until very recently coop games sucked. They were generally aimed at grade school kids with a theory about provided a way to play games that wasn’t competitive. That negative objective (not being competitive) didn’t really do much for game design and even grade school kids tended to find them boring.
This started to change when one of the first wave of rock start German game designers, Dr Reiner Knizia, built out a Lord of the Rings board game into a successful coop game in 2000. (Knizia was a contemporary of Tueber of Settlers of Catan fame. For reference, Catan was published in 1995 and became the first Eurogame that remains an international hit.)
Knizia did three things.
- The game was non zero sums. If any of the hobbits got to Mt Doom, all won the game. You could take one for the team and be a hero.
- He jacked the challenge so that you really did have to think not just about your own move but through a few rounds of moves or the game would eat your lunch. Even with excellent play the game would win 1/3-1/2 of the time.
- He used specialization, i.e. each hobbit had slightly different skills. The specialization wasn’t as radical as has become in later coop games but the bones of the structure had been established.
LOTR revealed a big advantage of coop games: no down time. Everyone had to be thinking all the time…even when it wasn’t your turn. And even if your hobbit got done in by the Nazgul, you could still kibitz and play a significant role. You were dead but not silenced.
There were further efforts in this direction, e.g. Shadows Over Camelot in 2005, some of which like SoC included a traitor so weren’t pure Coops. Then Matt Leacock designed Pandemic in 2007 and it was off to the races.
Here’s comments on some games written originally for cousin Kate to suggest games for middle schoolers and updated occasionally since.
In order of simplicity:

Mole Rats in Space – a coop reworking of Snakes and Ladders. Probably too simple. We played it 3 times. Got beat once. Haven’t thought to get it back out. (4 player – short game.)
No roles but charming playing pieces and an age appropriate challenge for 6-8 year olds and their parents.
Forbidden Island – locate treasures and get off the island before it sinks. You usually win in what feels like a cliff hanger. (4 player – 30 min)
Forbidden Desert.- find and assemble your airship before getting buried in sand. Will beat you relatively often. (4 player – 30 min)
Pandemic – the definitive Coop game. (4 players though you could squeeze in 5 – 60-90 minutes)

All 4 of these are by Matt Leacock with the last 3 using the same essential rule set (accelerating challenge, complimentary character roles) so could be introduced as a series with the Forbidden Island acquainting everyone with the core rules as complexity and difficulty increases.
The Captain is Dead. Stole the essential rules from Leacock but for up to 7 players with nice art and a Star Trek setting. (There’s a red shirt guy, the Crewman (image below), whose power is that if he or anyone else near him is going to be injured, he dies and reappears on the bridge.) This is, also, probably longer than 60 min but maybe you could stretch the game over a few days? One advantage is that it has a very nice mechanism for increasing the difficulty in so you can pump it in 7 steps from Coward to Insane. Which now that I think of it, also determines game length so maybe there’s a way to dial down the time hit.
Last: Pandemic Legacy – best game ever (ask Maggie.) Takes 12-24 session. The game board gets altered. Cards get torn up. Little boxes open with new pieces like an advent calendar. Appropriate if the above succeeds and you get some hard core coop gamers. In the BoardGameGeek top 10.
Just thought of one more: Flash Point (2008). In the Pandemic range of complexity. Same core rules. (Too bad Leacock couldn’t patent them somehow.) Good game for up to six players and claims to be under an hour in play time. I don’t remember it being shorter than Pandemic so you’d need to test that.
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Thought I’d add a few more for an older crew. This is not intended to be complete but only to report on games we’ve played a number of times.
Ghost Stories (2011) – same Leacock-esque core, great atmosphere (Taoist priests fighting ghosts for the sake of a village). It pushes the different roles out to different game tiles so depending on where your pawn is located you can activate different villagers to help. Same basic puzzle: how do we use a series of varying abilities to accomplish a tough joint task. Designer: Antoine Bauza.
Hanabi (2010) – another Bauza game and a huge favorite of ours. No roles but a tough joint task in a small box of cards playable pretty much anywhere. Takes 30 minutes to play and requires near telepathy to do well. Winner of the Spiel des Jahres in 2013.
Here’s an interview with Leacock and Bauza.
Mysterium (2015) – one ghost and the rest of the players are mediums helping the ghost discover its murderer. Was a bit light weight for us but very atmospheric. Has some serious fans on the Geek so it might be for you. It’s definitely the trippiest visually!