In the short story "Ripples in the Dirac Sea", you can experience going back in time, but upon returning, there is no effect on the current timeline other than your own memories of that experience.

I read some short stories about time travel (I thought by Niven, but I can't find descriptions of them online) which explicitly introduced the idea of "temporal inertia". If you changed the timeline and went back to your original time, reality would kind of slide towards the consequences of your changes; if you ran fast enough to your time machine, you might be able to go back again and fix it. That's a variation on open loop.

Primer is likely the most complicated time-travel movie, designed mostly to leave the viewer's head spinning at the end (like Memento, but more so). It's closed-loop, with some important restrictions on how time travel can work.

I have also seen the occasional use of perceived timeline alteration without time travel, such as when a character appears or disappears from everyone's memory due to mental manipulation. The universe still needs some restrictions on how easy it is to do that, but it's an interesting way to shake up the story without opening up the Pandora's box of paradoxes.