This is a bit different because typically, most of the people playing any given game have played it before, with only a few newcomers who learn the game as they play it. It wouldn't be a good way to play the meta-game, because you shouldn't expect to be competitive against someone who has played the game for a while.
It shouldn't matter as far as "learning to learn games goes", ideally. However, I expect it will, because you're not competing against other people learning to learn games. If you want to win -- and most people like to win -- the incentive here is to keep playing this one game until you're good at it.
I did something like this back in the 90s, with a guy in a local gaming club who had a big collection of boardgames, played a different one each time, and won about 50% of the time. It's important to bear in mind the typical boardgame does not have the depth of chess or Go -- those are classics for a reason -- and most of them are actually pretty much ideal for this sort of rapid pickup.
Earlier today I had an idea for a meta-game a group of people could play. It’d be ideal if you lived in an intentional community, or were at university with a games society, or somewhere with regular Less Wrong Meetups.
Each time you would find a new game. Each of you would then study the rules for half an hour and strategise, and then you’d play it, once. Afterwards, compare thoughts on strategies and meta-strategies. If you haven’t played Imperialism, try that. If you’ve never tried out Martin Gardner’s games, try them. If you’ve never played Phutball, give it a go.
It should help teach us to understand new situations quickly, look for workable exploits, accurately model other people, and compute Nash equilibrium. Obviously, be careful not to end up just spending your life playing games; the aim isn't to become good at playing games, it's to become good at learning to play games - hopefully including the great game of life.
However, it’s important that no-one in the group know the rules before hand, which makes finding the new games a little harder. On the plus side, it doesn’t matter that the games are well-balanced: if the world is mad, we should be looking for exploits in real life.
It could be really helpful if people who knew of good games to play gave suggestions. A name, possibly some formal specifications (number of players, average time of a game), and some way of accessing the rules. If you only have the rules in a text-file, rot13 them please, and likewise for any discussion of strategy.