Action selection games might ruin your idea, like in Race for the Galaxy, everyone picks actions and everyone gets to do all actions selected. Puerto Rico is similar, but actions are selected in turn order instead of simultaneously.
Dominion has the possession card, which gives player A control over player B’s turn, and player A gets all cards acquired by player B. There’s also cards that grant extra turns, with varying conditions.
Red Dragon Inn has a basic turn order, but every player have many cards that can be played anytime or in reaction to certain events. There’s also the gambling mechanic which gives everyone a turn in a mini game that doesn’t affect the main turns.