cooperative game that is NOT like pandemic, forbidden island et al
You have Sherlock Holmes Consulting Detective, in which the goal is to find the truth in the smallest number of steps. You can never lose to SHCD, you can simple have a very bad score.

In Shadows over Camelot, each quest will give you either white or black swords. The game ends when the players collected 12 swords or more and they win if more than half of them are white.

And my favourite: Les Mousquetaires du Roy (Ye King's Musketeers) is a semi-cooperative in which a player incarnates Milady, the evil spy of the Cardinal of Richelieu and all the others play Athos, Portos, Aramis, D'Artagnan and/or Tréville. Quests will reward the players but as they don't play against the game but against another player, the game isn't quite like a usual cooperative game.
The game can be a 1-vs-0 up to 5-vs-0 game, or a 1-vs-1 up to 5-vs-1 game.