While I think about the painting game described earlier, I can't help but think about a time travel game.
I've always wanted a time travel game where your control over time was far more flexible than it usually is. I don't mean the momentary time travel you find in Braid or Prince of Persia: I mean real time travel.
So here's the idea: the game is basically an adventure game. Much like any other adventure game, it features a large amount of running around getting items and solving puzzles.
UNlike any other adventure game out there, this one has a line running across the top of the screen, marked with logarithmic time (first inch is one minute, second inch is an hour, third inch is a day, etc). You are a dot in the center of the line, and the line slowly slides left past you. Actually, you're located just below the line, with a little line of your own behind you.
Here's an example of how the first scene might play out:
You arrive at the water park, but the gates are closed. The man behind the counter tells you that they open in about five minutes, if you want to wait. On the timeline, a glimmering gem is added. A helpful arrow signals that you should drag yourself to the glimmering gem. (The gem, if you mouseover it, says "water park opens".)
Dragging yourself up and over will move you to whatever time you drag to. However, this is not a "skip" - if you drag yourself five minutes forward in time, everyone else perceives you as having stood around for five minutes and you automatically fill in any events that happen locally that you could see.
So you move forward, go into the water park and, I don't know, steal a data disk. Then you decide to rewind time. Drag yourself up and to the left. Except that actually rewinds time: you're standing back outside the gate. The disk is in the water park, not your hands. You never took it. The line representing your personal timeline does not stretch forward and then back: as it always has, it starts from scene one and meets you wherever you are.
But there is an event on the timeline: "Hero steals data disk". If you were to simply drag forward, time would progress as it did before, and you would automatically steal the data disk. Because that's what you did. If you replayed the section, you would negate the event because you're doing something else.
However, that's dragging UP and to the side. There's also dragging UNDERNEATH and to the side (unlocked a bit later). This is an actual time skip, where you physically vanish and travel in time. This means there can be two of you, or maybe none of you. It makes your personal timeline either go loopy or get all dashed.
Your old selves cannot see your present self or feel your work, of course: the way the timeline works prevents that. Similarly, no matter how many of you are present, NPCs will only detect one of you. Unless interrupted, they will continue to perform the events set in their timeline.
So if you dragged DOWN and left after grabbing the disk, you would rewind to before the place opened. You wouldn't be standing outside the gate: you'd be standing wherever in the park you were standing, but it would be before the park opens.
You could then walk to the entrance, let it open, and then walk out and hand yourself the disk. The system looks forward in your original timeline, sees that you are slated to steal the disk, realizes that ain't gonna happen now, and collapses the time loop. You are now standing outside the freshly opened gate with a disk in your hand, having never time traveled.
Of course, you don't have to give yourself the disk. You could simply proceed forward without any issues. So long as you don't interfere with yourself getting the disk: this causes a paradox. Paradoxes are bad, and have to be resolved either manually (un-interfering) or by collapsing the loop as described above. Collapsing a paradoxed loop is rather unhealthy, though.
Most of the puzzles probably depend on there being more than one of you. Because that's fun and not hard to program.
This system can easily get very complex. For example, you might do a lot of looping in that area, and after a few hours of real play you'll find that your timeline looks more like sheet music than a line. Every event you are "slated" to do because you already did it appears on the line that did it, but it can still get unbearably complex. Collapsing loops can become a necessity, but it can also have unintended consequences, since all events you would have done in the loop aren't slated to get done anymore. They now never will automatically have happened, if you see what I mean.
(This is particularly bad if you travel a lot in your time loops: non-local events aren't displayed, and timelines that approach those events wander off the screen... Messy!)
You aren't the only time traveler, of course. Some items are "temporal", in which case they either stick to their time line (so you can't time skip with them) or stick to your time line (so you can rewind and keep them).
Some people are temporal as well, which means that if you time skip, they remember what you already will automatically do in your "old" time line.
A few people are even time travelers, although nowhere near as sophisticated as you. There will be a particularly interesting pair - one travels normal through time, the other travels backwards through time. They frequently give each other something to give back earlier in their personal time line (later in the recipient's time line).
All of these "chronally advantaged" people will have visible timelines of their own, of course...
Obviously, this "adventure game" features a bit more open-ended scripting than most. Think of it more as Quest for Glory or late-game ChronoTrigger.
That's just the time aspect... you could also make it so that you could "rotate" the axis so that instead of being time, it's parallel dimensions! Remember, traveling to alternate dimensions means you change to your alternate-dimension self...
Alternate version: as above, but it's a tactical game a'la Disgaea or Final Fantasy Tactics. Time hopping (going "below") can't bring your people back to life, but time scrolling (going "above") CAN... but at times you'll be forced to fight other time travelers at points in your career of their choosing...