In the Terminator series (movies and TV show), there are some odd time loops.
- John Connor sends Kyle Reese back in time. Kyle and Sarah have a son, John Connor. But John sent Kyle back in time only because of Skynet. Without Skynet, John wouldn't exist. The timeline protection hypothesis suggests John can't kill Skynet.
- Skynet sends a Terminator back in time. The Terminator's arm and CPU are left behind. The technology in that CPU is what Dyson uses to build the beginnings of Skynet. But Skynet sent the Terminator back in time only because of John Connor. Without John, Skynet wouldn't exist. The timeline protection hypothesis suggests that Skynet cannot destroy John.
How did we get into this circular timeline in the first place? I think it's reasonable for the initial timeline to exist without the loop. John could be someone else's son. Skynet could be developed without the Terminator's CPU. But once they start messing with time, they got into this circular dependency, where they only exist because of each other. I'm not sure they can get out of it though. It's similar to the grandfather paradox, except there are two parties trying to kill each other.
The Matrix series, which coincidentally also was about war between machines and humans, might give us a way out of the Terminator paradox. Agent Smith was trying to destroy Neo, and to do so he was willing to destroy the world. Neo sacrificed himself, which meant Smith no longer had a purpose, and Smith was destroyed at the same time as Neo. So perhaps John and Skynet have to destroy each other simultaneously. Or perhaps, as in The Matrix, the humans and machines call a truce, and both John and Skynet stop fighting far in the future, but only after the war that leads to both of them being created.