Complaining about games where there's no defining rule on whose turn it is to act reminded me of a game I learned last week, The Mind.
The Mind is completely about having no rule on when you can act; and I love it.
From the people who made The Game; basically everyone has a hand of cards, from a deck of cards numbered 1-100. The object is to play all cards from all hands in order, from lowest to highest, without any communication at all.
So when you think you have the next-lowest card, you play it. The whole game becomes about getting a feel for if you have waited long enough that you can be pretty sure that if a partner had a lower card, they would have played it by now. It's a lot of fun.
Reminds me of the really neat blue eyes / brown eyes island logic problem; where suddenly after 50 days every brown-eyed person will leave the island, because each brown-eyed person can deduce that if they had blue eyes instead, then the 49 brown-eyed people that they can see would have left on the 49th day.