I've had a few ideas, but never the drive to actually see them through.
- Backwards branching. Instead of deciding what happens next in the story, you're given a specific outcome or event to start the story and have to work backward to figure out how that result occurred.
- Time loops. You experience the same day continuously, with changes in the available choices appearing from time to time based on what choices you did or did not choose the "previous day".
- Multiple lives. Let's say you play as a knight; then you rewind and play as a sorceress, but it's during the same timeframe as the knight's part of the story and the actions the knight took affect the sorceress's story.
- Multiple perspectives. Similar, but instead of rewinding, you choose what perspective the next part of the story is told from. So really, the events that happen are the same, you just get to see the effect on different characters depending on what you choose.
- Flashbacks. A kind of puzzle, but basically the major events (besides the end) that happen are the same no matter what you choose (like the multiple perspectives one) and the difference is what information you find out. The main character is in a desperate/high-stakes/intense situation. Certain choices trigger different flashbacks, and the only real information about what's happening comes through those. You try to figure out what's going on with that info, and then use it to succeed or fail at the end of the story.
In a really similar vein to that, dream pieces; a series of fragmented dreams or memories are shown as the story progresses. You choose which dream or memory to explore, with each choice revealing a different piece of the MC’s past or subconscious. It all may seem disconnected at first, but they build toward a significant revelation about the character’s psyche or the overall mystery.
Some of these may really just be fancy takes on cave of time structure, and I definitely think a few of them would end up just seeming very linear if not handled correctly. I think with the proper amount of effort, though, they'd be pretty interesting.