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I think some card games meet your criteria. Specifically, Bridge, Hearts, and Auction Pitch (aka Setback) are all complex, turn-based, imperfect information games with some amount of cooperation. I’d love to see DeepMind take on one of these games.

Competitive Pokemon with teambuilding would be something fresh I don't think existing techniques have figured out yet.

Not super prestigious, but IMO still a good intermediate goal.

Civilization?

AlphaCiv would be interesting as hell, but I fear that there isn't historical data for Alpha to consume. I would expect it would have a similar result to AlphaStar though-- it beats the human player before the end game.

Perhaps it should have other targets than simply beating the opponent. Like getting good statistics.