In Black Maria the protagonist and her mother go back in time while turned into cats to watch events unfold, only to become responsible for the events in the first place.
In The Merlin Conspiracy, there is a doozy: Nick's part of the story begins with him sidestepping into other universes willy-nilly. A man named Romanov shows up to kill him. After he spares Nick's life, Nick later follows him to his personal island/mini-verse, where he is deathly ill. While taking care of Romanov, a Knight Templar and his two wards (Joel and Japeth) show up to finish Romanov off while he's down. They are dispatched, and one of the boys treads on an egg. Nick laughs. Several universe hops later and Nick winds up in Blest, where there is a ten year difference between Romanov's world, and the two boys are now the Big BadChessmasters who kicked off the whole plot to begin with, and are the ones who not only sent Romanov to kill Nick, but also gave Nick a special virus to kill Romanov, which was what made him sick earlier. Which Nick himself accidentally gave them the idea to do. He accused them on Romanov's island -while they were still young- of paying Romanov to kill him. Plus several other, minor time-related things, such as answering the phone while Romanov was ill, causing wife to leave him. She winds up being Joel and Japeth's sidekick.
There was a short sci-fi story where a king, who is always coming up with crazy and unhelpful schemes to improve his small country discovers that a time traveler is helping his advisers to offset the impact of his schemes. He captures the time traveler and forces him to take them both into the future so he can see how things will turn out. They arrive in ten years in the future and that the country is prosperous beyond his wildest dreams, so he asks a passerby "What was the big change that brought about this golden age?". The passerby answers "It all turned around when the crazy king disappeared ten years ago and the advisers started ruling the nation". As the king wonders why he disappeared a decade ago, the time traveler shuts the door to his time machine, leaving the king in the future.
In Douglas Adams' Life, the Universe and Everything, the poet Lallafa was known for writing beautiful poetry on habra leaves in the middle of a rainforest... So some time travelers picked him up from the rainforest and put him on the talk show circuit in the future. Of course, he had to write the poems at some point, so they just sent him back to the forest with a book of his poetry and a bunch of habra leaves...
Far from the only one in the series. Arthur, of course, met with Agrajag before one of the many deaths of his previous forms had ever occurred, and so he knows that he's going to be able to escape when Agrajag tries to kill him anyway. Also, the entire arc with the Golgafrinchams.
As explained by Ford Prefect, every form of Time Travel in that universe is a Stable Time Loop.
Zaphod Beeblebrox is his own ancestor and descendant.
That series of books is definitely Timey-Wimey Ball. People are trying to build an ion factory. They don't finish it in time so they keep pushing the construction start date back farther into the past, until the cathedral that was originally in the spot was never built in the first place. It then states that photographs of the cathedral suddenly became immensely valuable. Huh? Time travel, like everything else in those books, runs on Rule of Funny.
What's more, in the first book, it's suggested that the origin of life was caused by the Infinite Improbability Drive — which was, of course, later built by living creatures.
In the story The Red Queen's Race by Isaac Asimov, an attempt to change history by sending modern scientific knowledge back to the ancient Greeks is subverted when the person translating the information finds out about the plan. The translator creates one by censoring the translation to include only odd bits of surprisingly advanced knowledge that actually turned up in the ancient world (according to him, it wouldn't have mattered that much in any case - ancient Greek mentality has little use for applied sciences). Also, it's decided that doing this was necessary for history to happen as it already did.
Also implied in "The Last Question" ("Can entropy be reversed?") when the Cosmic AC has finally compiled enough data to come up with an answer (despite the end of the Universe) and says LET THERE BE LIGHT!
Not to mention they go back and close down each time loop so that they never actually happened, leaving the protagonist deeply confused as to how he/she still exists. The person she's with tells her basically to shut up and deal with it.
The entire plot of Artemis Fowl: The Time Paradox. A picture of Artemis chasing himself through the timestream is on the cover. In going back to the past to rescue the lemur, Artemis managed to draw the attention of the younger Opal Koboi. She then followed them to the future, but arrived a few days early, in order to set off the events that would cause them to time-travel in the first place.
Artemis even lampshades the trope early on, before any actual time-travelling takes place:
Plus, Artemis' adventures in the past are implied to inspire his younger self to research the faeries and begin exploiting them in his schemes.
Lester del Rey's 1951 short story "...And It Comes Out Here" features a time machine that's created by a time loop: the protagonist is given it by a future version of himself, who was given it by a future version of himself, and so on, so that he can do what his future self remembers him doing. The story also involves another ontologically paradoxical object: The protagonist travels into his own future and steals a prototype invention from a museum display which credits him as the inventor, then takes it home and passes it off as his own work.
In the Thursday Next Novel The Eyre Affair, Thursday meets herself, and receives the news that the Big Bad is alive, and is told to travel to Swindon. As a result of the travel, she ends up caught in a patch of Bad Time, and arrives to deliver the message.
Later in the series, it's revealed that the various methods of Time Travel work on the assumption that someone will invent Time Travel, and deliver that technology to their current time. This starts causing trouble when people find that Time Travelwon't be invented.
Also, Thursday's father gives her his chronometer. He got it years ago from her, after she got it from him.
A debate runs throughout the book about who really wrote William Shakespeare's plays. At the end of the book, Thursday's dad, a time traveler, reveals that no-one ever wrote the plays; when he went back in time to the corresponding period, the plays weren't around. So, he gave them to William Shakespeare to produce. Thursday's dad tells her not to worry about where the plays actually came from, as these things happen often.
In Something Rotten, Thursday meets her father when he turns up to thwart and investigate an assassination attempt on her. He's there because of a conversation they had "three hours ago", and refuses to answer Thursday's questions because he's already explained all that and can't be bothered to go over it again. It turns out that through a minor confusion (the aforementioned chronometer is broken, so Thursday gives him "hers"), the previous meeting is in fact three hours in the future from Thursday's point of view. Three hours later, Thursday happens to be passing her father's office and decides to go in to find out what he'd been talking about earlier. At the end of the conversation (which is equally as confusing for her father as the earlier chat had been for Thursday) she mentions that she was only there because of their talk during the assassination attempt, and her father agrees that perhaps he should go and investigate...
The Robert L. Forward novel Timemaster demonstrates the use of a Stable Time Loop generated by a wormhole (technically, a "closed timelike curve") as an offensive weapon.
The protagonist of The Stainless Steel Rat gets dumped into Stable Time Loops so often that he treats it as a normal occurrence. Usually they're fairly brief, but The Stainless Steel Rat Saves The World is a novel-length stable time loop. Or more accurately, it's one stable time loop after another for the whole book, with one unstable time loop for variety.
The Technicolor Time Machine hinges on several Stable Time Loops. The premise is that a movie studio is about to go bankrupt, and so in desperation they try funding a seemingly crackpot physicist who's working on a time machine in exchange for the use of the completed model. It works, of course, so they take a camera crew back in time to film a historical about how the Vikings discovered America — they don't have to pay for sets or actors this way, and they can get the whole film done in a couple of days so they'll be able to show the bank that they have an asset they can monetize when the next loan payment becomes due. When they find the Viking that history says is the discoverer, however, he seems completely uninterested in attempting the journey... until they nudge him with a little bribery and technical assistance. A few other Self-Fulfilling Prophecies occur later on, including a note that nobody wrote and a vicious practical joke one of the characters plays on himself in revenge for that same vicious practical joke he played on himself 'earlier.'
Actually, they don't know that this Viking is the discoverer. He's simply the first one they encountered, so they decide to go with him, especially since he's willing to work for crates of Jack Daniels. Another time loop happens when a ship arrives for him, which the seller claims that he bought. This clues the time travelers that they need to go back and buy the ship. They have the Viking sail for Vinland (the New World) and then time travel to Vinland to film the ship arriving. After the filming is done, they start wondering if they'd just changed the past by having someone else other than Thorfinn Karlsefni lead the expedition. The Viking reveals that this is his real name. The time travelers realize what they have done by accident. Also, the director is told later that the person who supplied Thorfinn with the ship was Bjarni Herjólfsson, which sounds like a bastardized name of the director.
"The Secret of Stonehenge" is a short story which features a group of scientists setting a camera on a time machine in the middle of Stonehenge and sending it back to the estimated time of creation and then sending it on a series of short jaunts forward in time to capture what had been going on. After the machine disappears, they notice that sending it that far back in time left a ghostly image for a moment. Moments later, they realize the reason Stonehenge was built... in tribute to the ghostly thing that kept appearing in that location over the years...
The Robert A. Heinlein short story —All You Zombies— uses the same device. The protagonist tells a bartender a story in which he introduces his mother, actually himself before a sex change, to his father, actually himself after the sex change. He is also the bartender, sent back in time to recruit himself into the time-travel police.
In The Door into Summer, the protagonist travels into the future and sees machines he's almost sure he invented. So on that hunch, he finds a time-machine that can send him back. He makes some arrangements, returns to the future by cold sleep and lives happily ever after knowing the people who tried to ruin his life got their just desserts.
James Hogan uses a Stable Time Loop approximately 50,000 years long in the third book in the Giants series.
Time Travel in the Dragonriders of Pern books by Anne McCaffrey operates on this principle. Much of the Time Travel is undertaken knowing in advance that it will work ("since I've already done it, I might as well go do it..."). What isn't, is of the form "I think I'm the one who did it, so I'd better go do it..."
Behold the Man by Michael Moorcock: Karl Glogauer tracks down the real Jesus, son of Mary, and finds that he's an idiot; so he...
The Time Traveler's Wife by Audrey Niffenegger is full of Stable Time Loops. For example, on one of Henry's visits to his wife-to-be Clare in the past, he dictates to her a list of dates in her childhood when they're going to meet. They meet on those dates only because she knows he's going to appear — there are some other times when he appears, but since those dates aren't on the list, she doesn't know he's there. When they meet as adults in real time, she gives the list back to him so he can memorise it. Where did it come from in the first place? Seemingly nowhere. Henry also taught his younger self a number of skills he knew he would need, such as how to pick locks. His theory is that to prevent Temporal Paradox, he has free will while he's living in normal time but not while he's time traveling.
One time loop is Henry and Clare's marriage. From Clare's perspective, she meets Henry when he travels back in time to her childhood, lands in her backyard, and introduces himself as her future husband. From Henry's perspective, he meets Clare when he runs into her in a college library and she tells him that she's known future-him for most of her life and that they're going to get married. So when did they meet for the first time? The answer is "Both": He first met her at the college library, and she first met him in the Meadow. Because of Time Travel, they both met a version of the other who was ignorant of their future relationship/marriage.
In H. Beam Piper's short story Flight from Tomorrow, a tyrant in the very far future forces a scientist to create a time machine for him as the ultimate escape route, and he uses it to flee into the past from a rebellion at the beginning of the story. He is not expecting a Stable Time Loop — quite the contrary — but the scientist not only left out some important information but sabotaged the machine, so that he went back not to the time he had researched, but to the mid-twentieth century. The tyrant is hunted down and killed as a plague-carrier. The scientist in his own time explains to the rebels that they cannot pursue the tyrant into the past, or they will meet the same fate; the scientist's audience realizes that a mysterious artifact from the distant past must be where the tyrant's body was covered over with concrete to prevent further contamination.
In The Anubis Gates by Tim Powers, the protagonist, Brendan Doyle, becomes the victim of Grand Theft Me in the 1800s and realizes that he is destined to be the poet, William Ashbless, whom he was researching in the present day. Partway through the book, he panics on realizing nobody ever wrote Ashbless's poetry — he copied it from memory earlier — but then shrugs it off, deciding that as long as it was there, nobody would be bothered.
The Discworld novel The Last Continent is essentially a single, but quite complex, one, in which the problem Rincewind has to solve is caused by the wizards accidentally going back in time while looking for him. It also includes Ridcully dismissing Ponder Stibbons' worries about the Butterfly of Doom (or Ant Of Doom in Ponder's example) by concluding that history depends on you treading on the ants you've already trodden on.
Specifically, Ridcully's argument relies on the old "you can't step on an ant if you don't exist." His logic is that if they're in the past NOW, then they've already been there thousands of years ago, when it was now. Therefore, anything they do, they've already done (because it's the past and the past has already happened), and it's vitally important that they do whatever they do, because if they didn't, they wouldn't have done it and they'd have done the different thing instead.
Night Watch subverts a Stable Time Loop: there was a real Sergeant Keel the first time around, but Vimes' and Carcer's arrival from the future gets him killed ahead of schedule. Vimes must assume Keel's role to force stability on the Loop, and while the general outcome is the same, several of the specific events are different.
On a smaller scale, minor recurring character Mrs. Cake is a psychic who is known to answer peoples' questions before they ask them; she then insists they ask, to stabilize the time loop, or she'll get a migraine.
In Interesting Times, Hex answers a problem before it is asked. The wizard in charge eventually enters the problem to appease causality, but not before hiding in the privy for an hour and a half.
In Eric, Rincewind travels back in time to before life existed on the Discworld, and drops a partially eaten sandwich in a tidepool. The microorganisms in the tidepool become the ancestors of all life on the Discworld, including Rincewind (but not including the sandwich ingredients, because the sandwich didn't originate from Discworld; it was given to Rincewind by the creator of the universe).
Played with extensively in Pyramids, particularly in the construction-crew's "doppelgangs" and Dios's fate. The paradoxes entailed are lampshaded when the engineers discuss the option of paying their loop-duplicated workers with loop-duplicated money.
In Soul Music, Susan travels to the past and sees her father fight Death at the conclusion of Mort. Death spots her watching and recognizes her as the child of Mort and Ysabell, which convinces The Grim Reaper to stabilize the loop and spare his apprentice so the girl he's just spotted can be born.
Also the first paragraph of the book takes place later in the story, despite being chronologically first.
In I Shall Wear Midnight, elderly Tiffany insists this trope is not in effect, as each iteration of this time-traveller's encounter with young Tiffany will actually result in a different conversation. The fact of their encounter is stable, but the details aren't set in stone.
Harry Potter and the Prisoner of Azkaban. Harry is saved from dementors by a Patronus Charm cast by a mysterious figure who he thinks is his father. After he travels back, he eventually finds himself in the same place and waits for his father to show up... and then realizes HE was the mysterious figure, and saves himself. In fact, he only gains the ability to cast a true Patronus for the first time because he realized that he had already done it. Also, as Harry, Ron, and Hermione first head out to adventure, they hear noises that turn out to be Harry and Hermione as they complete adventure!part I.
This is often used as a justification for being unable to change history in the Potterverse, but it seems to contradict what Hermione tells Harry in Prisoner of Azkaban about wizards and witches having to be careful to avoid killing their past and future selves. One can explain this in various ways, but the end result is that canon is not entirely clear.
Possibly subverted in the Star Trek: Deep Space NineExpanded Universe trilogy Millennium, which involves a convoluted Quasi-Stable Time Loop in which the actions of a future Picard and Nog (as well as a present-day Vash) help cause the creation of their alternate future, followed by the retroactive destruction of that same future. During the story both Dax and Miles O'Brien continually insist on maintaining one, but by the end it seems their actions can only succeed because of two people who shouldn't exist (and a third which shouldn't even have been there), as well as the presence of a futuristic ship from the same Bad Future. The epilogue refers to the resulting double time loop as "circles wheeling within circles".
In the Warhammer 40,000 novel Desert Raiders, a Tallarn regiment is dispatched to an uninhabited planet to investigate a mysterious psychic distress call. After landing on the planet, the regiment encounters a Tyranid splinter group and is forced into a desperate last stand. One of the psykers traveling with the regiment dispatches a warning signal in their final moments — the same signal the regiment had been sent to investigate in the first place. The implication is that, in traveling through the Warp, they had gone back in time before reaching their destination; indeed, the Warp in the 40K 'verse is known to do some strange things to the flow of time...
In Animorphs, In the Time of Dinosaurs, the Animorphs go back in time to the Cretaceous, fight the antlike alien Nesk for a nuke to explode (so that they can undo the time travel) and the Nesk divert a comet to the only home of the Mercora (the friendly aliens). The Mercora wanted the nuke so that they can explode and stop the comet from hitting, but Tobias and Ax rig the nuke not to explode, as the comet was the one that ended the dinosaurs (opening the way for humans to evolve). The force of the comet ends up sending the Animorphs back home.
In #13, The Change, the Ellimist sends Tobias back to the night before the infamous construction site incident that started the whole series. Tobias thus convinces his past self to team up the next evening with Jake and Marco and then to take that shortcut across the shady construction site.
Also, the whole Elfangor's Secret book.
Dragonlance Legends reveals that humans, elves, and ogres can time-travel only to observe. This is how it's supposed to work. Throw in the unnatural races, which were not created at the beginning of time, like dwarves, gnomes, and kender, and you have problems. So, Raistlin during the Twins Trilogy would be caught in a stable time loop which essentially just causes him to kill himself over and over again every 400-odd years... if it weren't for Tasselhoff's (a kender) jumping into the spell at the last minute and mixing everything up.
Played with in Kurt Vonnegut's Slaughterhouse-Five where the character lives in a personal unending non-chronological time loop where he lives out every moment of his life repeatedly, with all of his own memories, after becoming Unstuck in Time. In the novel he is suggested to have lived out all these moments more than once and always does the same things every time making it a stable time loop of sorts...
One book continuity of Red Dwarf or the other puts Lister as the creator of the universe, having gone back to that point in time to see what happened.
The William Tenn short story "The Discovery of Morniel Mathaway" is built around a stable time loop that involves an art historian meeting the object of his research.
The short story entitled The Sky Looked Strange Today involves a man driving on the freeway when he is distracted by a peculiar aurora on the horizon. When he looks back at the road, he crashes into a taxi cab that has started a pile-up, but time slows down for everything but the man. He gets out of his car into a motionless world when suddenly, his "guardian angel" appears and tells him that his next door neighbor caused the crash. He takes the man back in time, one hour earlier, and leaves him to figure out how to stop the pile up. In an impulse effort, the man slashes his neighbor's tires and, feeling he has succeeded, takes a cab to work before his past self sees him. As he rides in the cab, he notices the aurora again. Unfortunately, so does the driver. The man forgot that his neighbor works for the cab service. So, the neighbor rear-ends a car, causes the pile up, and just before the man is killed by his past self (still distracted), he says his odd last words: "The sky looked strange today..."
To Say Nothing of the Dog is built entirely around this trope. The main characters spend virtually the entire length of the novel time traveling back and forth to the Victorian era, trying to correct the actions of one of them that threatens to change the entire course of history.
Used quite effectively in Simon R. Green's Deathstalker series. Owen Deathstalker and his companions receive superpowers by passing through the Madness Maze, an alien artifact built to combat a terrible menace that the aliens knew about. After the protagonist dies his true love, Hazel D'Ark, is driven insane by grief and resolves to go back in time and become so powerful that she can prevent it from happening. It turns out that she is the horrible unknowable menace the Madness Maze was originally designed to fight.
In Andrey Livadny's Ark, all the worlds encountered by the main character turn out to be biospheres built into the titular Ark for the various alien species on-board, an enormous Generation Ship literally built out of the Moon by humans thousands of years before in order to basically follow the Star Trek mantra. Most of the logs are lost, and the ship's AI has no idea where they are or even what year it is. Without the crew to aid in maintenance, the Ark is in a dire state of disrepair. They manage to find a yellow dwarf star nearby with a habitable planet. Since the spherical craft was never meant to land (imagine the tidal forces from a Moon-sized object), they are forced to drop it in water in hopes of cushioning the impact. They do as much as they can to brake before hitting the atmosphere. The main character, who is now an electronic consciousness in the ship's computer, separates the command module from the rest of the ship and lets it fly away from the planet with himself and the ship's AI still in it. The Ark somehow manages not to break apart on impact, although it creates massive tsunamis and empties out the sea they hit. Most of those on-board survive (probably due to some sort of Inertial Dampening). One of the first people to get out is an old shepherd who introduces himself as Noah. The novel ends with the protagonist returning to the planet after several thousand years and teaching the inhabitants several important values, including "Thou shalt not kill."
Time travel in Poul Anderson's novel There Will Be Time seems to require these.
In "Explain the Internet to a 19th Century Street Urchin", from the book ''Everything Explained Through Flowcharts", these are some of the more favorable outcomes. The non-temporal outcomes usually result in your death.
The short story collection Short Trips: Time Signature follows a single piece of Vortex-threatening music through the Doctor's life. Since the book is in extreme Anachronic Order, following neither the music nor the Doctor linearly, it takes a bit of working out, but essentially the music was sent to the planet where the Doctor first heard it by someone who'd heard it from the Doctor. ( But then the same person helps the Doctor disrupt a key point in the chain, so none of it happens after all. Wibbly-wobbly timey-wimey.)
The events of the novel The Stone Rose begin because Mickey sees a statue of Rose in the British Museum. By the end of the story the statue still hasn't been made, so the Doctor carves it himself. In the same book, a vial of mysterious liquid turns out to have been created by the Doctor, by running the dregs of the vial through the TARDIS's Matter Replicator, and taking the result back in time.
The Fourth Doctor short story "Breadcrumbs" has the Doctor going to a lot of trouble to collect various three-dimensional data fragments and run them through a reversed Matter Disperser in order to find out what they are. The last fragment is very close to a wormhole, and he just has time to register that the "message" is another Fourth Doctor, before he gets sucked into the Vortex, and ends up on a deserted island on an unknown planet. His only hope is to rewire the Matter Disperser correctly, and transmit himself as a series of three-dimensional data fragments...
The short story collection Decalog 2: Consequences has the gimmick that the Doctor's actions in one story trigger the events of the next story. And, of course, the events of the last story trigger the events of the first.
In the collection Short Trips: The Centenarian, about how one ordinary but long-lived man keeps meeting the Doctor, the final story in which Edward is 100, takes place at the same time as the first story, in which he's born, and it's suggested that it was the centenarian Edward's actions that linked his life to the Doctor in the first place.
The Framing Story of Short Trips: Defining Patterns has the Sixth Doctor go back to 1957 to find out why certain material was removed from UNIT files. Checking what the files actually say, he concludes that they're too dangerous to stay in human hands, and removes them.
The short story "Tautology", published in Doctor Who Magazine #194, reveals that K9 is the result of a stable time loop. Professor Marius is inspired to build K9 Mark I after examining the millennia-old remains of K9 Mark III, who was built and given to Sarah Jane Smith after the Doctor had met K9 Mark I.
There and Back Again by Pat Murphy has a mysterious note in the protagonist's handwriting that appears at the beginning of the book, and a mysterious message asking for help that gets a number of secondary characters to arrive at the climax of the book in time. Both turn out to have been sent by the protagonist himself at the very end of the book, after he goes back in time.
A common theme in the Xeelee Sequence books by Stephen Baxter. The Xeelee, who first evolved who-knows-when-and-where, reach back in time to refound their civilization at the very beginning of the universe. This gives them a multi-billion-year technological jump, which they'll need to fight a war which would have destroyed them long before they could reach back in time. It's actually to escape the universe— the war was already lost and was always unwinnable.
In The Dandelion Girl, a man whose wife is away encounters a girl in her 20s coming out of the woods who looks somewhat familiar and wearing a strange-looking dress. She talks to him a little and then goes back to the woods. She proceeds to return several times and, eventually, tells him that she is from the future. Time travel is a possibility where she's from, but the government has banned it for fear of changing the past. Her father secretly built his own time machine before his death, as he believed that time is immutable and everything has already happened. In the meantime, the man's wife starts to act a little strange towards him, as if she suspects he is spending time with another woman. The girl is missing for several days, and then comes back saying that this is her last visit, as the time machine is about to break down due to lack of maintenance. There may be enough left in it for one more trip. As she disappears into the woods, he follows her but sees only a bright flash of light. He returns to the house and looks in the attic for something, only to find his wife's old things, which also include the same strange dress the girl wore, the same dress his wife wore on the day they met years ago. Everything suddenly clicks in his head, and he realizes he had been married to her all along.
Warrior Cats also features one. In Starlight, the cats find a perfect (and uninhabited by other cats) spot to live. If there had been other cats, they wouldn't have been able to stay there. Later, in Long Shadows, Jayfeather travels into the past and convinces the cats living there to leave for the mountains, which he could not have done had he not lived there. Then, Rock appears and tells Jayfeather that he remembers that Jay's Wing (the cat everyone mistook Jayfeather for) disappeared after the cats left for the mountains. Because of this, he takes Jayfeather back to his own time, causing his memories of Jay's Wing's disappearance. Also, in Outcast, Jayfeather met the Tribe Of Rushing Water, and learned their customs. Then, in Sign Of The Moon, he travels back to the Ancient Cats and teaches them the Tribe's customs, allowing them to become the Tribe.
The entire plot of When You Reach Me is Miranda slowly discovering that she is in the middle of one of these.
Sergey Lukyanenko and Yuliy Burkin's novel Today, Mom! has two brothers being taught Ancient Egyptian by their archaeologist mom. They end up discovering a time machine inside an ancient artifact and traveling to the future and then the past. After saving a young girl from being married to a dying pharaoh (she would be buried alive with him), they take her to the 20th century and leave her there to start a new life. When they get back to their own time (1993), they are attacked by the mummy of said pharaoh who has come back to life. Suddenly, their mom bursts into the room and destroys the mummy with a weapon from the future. The boys realize that the girl they saved is their mom who knew what she was doing by teaching them Ancient Egyptian.
A millennia-long time loop is central to the plot of Aleksandr Zarevin's Lonely Gods of the Universe, although it's not revealed until the second half of the book. Of course, the characters realize that the time loop is far from stable and will inevitably collapse after 5 or 6 cycles (what that means is anybody's guess), destroying everyone and everything in it. They spend the rest of the novel trying to break out of the time loop, namely by preventing their births, while making sure that their present selves stay alive. Let's just say the temporal mechanics get very confusing by the end.
A smaller time loop occurs in the middle of the novel. The main character is asleep in his apartment, when he hears a loud thud in the next room. He finds a strange object that appears to have been neatly sliced diagonally. Not sure what to make of it, he throws it in the back of his closet and forgets about it. A year later, him and his friend are experimenting with a strange machine they built, which appears to be a mix of a teleporter/portal. One of the tests is a long object specifically made for this. In the middle of the test, power cuts out, resulting in a Portal Cut. However, the part of the object that went through is nowhere to be seen. The main character quickly runs out and brings back the other part from his closet, explaining what must have happened. Eventually, they get additional funding and turn it into a Time Machine.
Additionally, by preventing their own births, the Human Aliens are also saving their own civilization from a nuclear war, which resulted from one of them creating a teleporter (which they used to get to Earth).
Comes up in Dinoverse. Bones from a large dinosaur closely related to Tyrannosaurus rex were found at a site called the Standing Stones, a series of, well, standing stones. Bertram puts a shard of one of those bones in his science fair project, which accidentally turns out to be a time machine that sends his mind, and those of some people around him, into the past and into the bodies of dinosaurs and a large pterosaur. They get a message from the distant future, sixty years after the time machine came on, telling them that their bodies had been in comas for sixty years but someone had found a way to fix things. If they could just get to the site of the Standing Stones, they could go back. On the way there is discussion as to whether they'd already failed, and when they got to the place there were no stones, plus they got attacked by a large Tyrannosaur and barely managed to kill it. Turns out it was the same dinosaur, and they had to set up the stones themselves before they could go back.
Count and Countess makes use of this frequently. The two main characters, Vlad Dracula and Elizabeth Bathory, have been writing letters to each other across time since they were young children, and often what one character writes to the other will have a large impact on the recipient's timeline, depending on how he/she acts on it.
In Barrington J Bayley's novel "The Fall of Chronopolis", a gay man called Narcis travels back five years to seduce his younger self away from the latter's boyfriend. Five years later, they realise someone will soon arrive to destroy their relationship. The book doesn't explain how the traveller didn't remember this had already happened.
Toward the end of The Secrets of the Immortal Nicholas Flamel series, the conflict becomes less about preventing bad things from happening and more about traveling back in time to ensure things happened the way they did.
In the Elephant & Piggie book "We Are in a Book!", Gerald panics near the end because he doesn't want the reader to finish. Piggie solves the problem by asking the reader to read the book again... which begins with Gerald whispering to Piggie, "I think your plan worked."
In the adolescent novel The Eyewitness by Caroline MacDonald, an eleven year-old boy inadvertently travels forward in time and befriends a teenager whose grandfather committed violent suicide several years earlier. At some point in his life, the time traveler realises that he was that man, and when the time comes around, he kills himself because it is his fate to do so.
The short story The Janus Equation has a bizarre version: A man is seduced by and has sex with a mysterious woman who disappears after becoming pregnant. It turns out that the woman was himself travelling back in time after a sex change. First there's the obvious loop of him getting the sex change and travelling back to do that in the first place, but then Fridge Logic kicks in and you realize that since his/her/their child was made from one person's genetic material, will that child actually become him/her/them? Not necessarily — twins aren't clones — but you never know with time travel.
In Noob, the Galamadriabuyak tower transports players in random places in the past to have a Boss Fight. All these fights are stated to be this trope, as the game world's history will not change as long as the boss is killed.
In Theirs Not to Reason Why The Immortal was born two centuries in the protagonist's future, then cast back in time 15,000 years by the Feyori. She rescued a group of humans from tectonic upheavals on Earth and transported them to a distant planet, which eventually became the V'Dan Empire, and founded a religion there predicting Ia's coming as the Prophet of a Thousand Years. This also allows Ia to steal designs for future technology from the Immortal's records.
The exploits of Thot Tran in the Star Trek Novel Verse. His attempted recovery of a crashed Alternate Universe starship from Tirana III in Star Trek: Cold Equations is a failure, costing the Breen vast amounts of money and resources, as well as embarrassing them politically. This failure leads to his fellow Breen Thot Trom penetrating the alternate universe in Star Trek: Section 31 - Disavowed - where he ends up hijacking a starship and being sent several years into the past, crashing the ship on Tirana III. Realizing in his last moments that he is responsible for the entire wasteful affair in every way, Trom spends those moments laughing crazily. As for Tran, he defects to an allied nation.
Due to the ease of time travel in the setting, Greg Egan's Orthogonal trilogy predictably features a few standard loops, but because of the nature of spacetime (time is fundamentally no different from the dimensions of space, and spacetime is a Wrap Around, probably in the shape of a four-dimensional torus or sphere), the entire universe actually exists as a massive Stable Spacetime Loop. One of the more interesting implications of this is that not only does the universe periodically have a Big Crunch followed by a Big Bang, but every time it happens, it's actually literally the same event.
Partially deconstructed in the first book, The Clockwork Rocket, when a character wonders if free will is just an illusion in such a universe, and Yalda explains her opinion of why it's not: The loop exists on such a beyond-astronomically vast scale that an individual's actions are almost insignificant in comparison. More importantly, it's (mostly) impossible to see the predetermined future, so the question of free will in a predetermined Stable Time Loop is a moot point. On the other hand, legitimately predicting the future and traveling through time does become practical later on in the trilogy, and the ramifications are explored.
Fully deconstructed in The Arrows of Time. The main plot is driven by the idea of a messaging system that will allow people to send messages back in time. Cue extensive discussions about why this could not possibly lead to a Temporal Paradox... but it does cause a serious case of Space Age Stasis.
At one point in The Arrows of Time, while en route to a planet that is essentially traveling backward through time, a few travelers have an interesting conversation about leaving a spyglass on the surface to decay: From the planet's "perspective", it will be as if a rock eroded into sand, which then gradually and autonomously formed itself into a spyglass, which would then be picked up by the visitors and taken off with them into space. Things get weirder when someone wonders what they may have on the ship with them that came from the planet that nobody has ever set foot on yet—and they get even weirder when they realize they've had planetary dust on their ship all along.
In Stephen Baxter's The Time Ships, the authorized sequel to H. G. Wells' The Time Machine, the Time Traveler explains early in the novel that his time travel research begin after a mysterious stranger arrived at his door one day and gave him a piece of material that he used to construct the Time Machine. The stranger turns out to be none other than the Time Traveler's future self, arriving from the future to ensure his ability to travel one last time through time to rescue Weena.
In Scott Meyer's Magic 2.0 books, Brit the Younger and Brit the Elder are the same person, but Brit the Elder is Brit the Younger's future version who has/will travel back in time to build the city of Atlantis, where they live now. Brit the Younger is frequently annoyed and miserable thanks to Brit the Elder treating her like a child and concealing crucial information until after it would have been useful. Why? Because that's how Brit the Elder remembers it from when she was Brit the Younger. Both Future Me Scares Me and I Hate Past Me are invoked here. Phillip is a big believer in free will and is adamantly opposed to the very idea of a Stable Time Loop, which is one of the reasons why he and Brit the Younger hit it off. He keeps insisting that Brit the Elder is simply a projection of Brit the Younger created by the reality program as a result of her actions. Brit the Younger gets really annoyed with Phillip when he acts "familiar" with Brit the Elder after spending the night with Brit the Younger. He points out that she can't really feel jealous of herself, to which Brit the Younger replies that he himself doesn't believe that Brit the Elder is really her. They do, however, look nearly identical (since time travelers are The Ageless) except for their outfits.
In Captain Underpants and the Revolting Revenge of the Radioactive Robo-Boxers, George & Harold along with the two Tippys, Sulu, and Crackers battle in various different time eras as they attempt to battle each other. The two Tippys use a nuclear bomb that kills the dinosaurs, George & Harold train some cavemen to paint on walls which invents comics, Tippy uses his freeze ray which cuases the ice age, and at the climax of the story, the boys' pets destroy the Tippys by blowing up an atomic bomb in a time even before the Universe existed, which in turn causes the Big Bang and forms the Universe.
Tanya Huff's The Enchantment Emporium lists a particular story as how the Gale Family got started — a "horned God" with a woman that would become the first Gale woman. Then, in Falling Skies, it turns out that one of the main characters, in trying to use Time Travel to bring back one of her cousin's sons to solve the main problem of the book, actually needs to take him back (along with another relative) to become the first Gale's. (And a different character is brought back to 'the current day' to actually solve the problem; it turns out the older version had already shown up several times in the story, but they didn't recognize him — he's a shape shifter and the older version only used his alt-form.)
"Transfer Point" by Anthony Boucher has a man from the far future discover, and avidly read, the science-fiction pulp magazines from the early 20th Century. He is struck by the fact that one particular future-history series, with the byline "Norbert Holt", is amazingly accurate in its predictions, even down to specific names and dates. He then accidentally gets sent back in time to the pulp era, and to earn a living starts writing pulp fiction — under the byline "Norbert Holt". It is hinted at one point that he is aware, and his editor vaguely aware, that this is not the first time around the loop...
This trope is the main plot hinge of A. E. van Vogt's "Not The First!". To say any more would be a spoiler.
Likewise, this is one of the plot hinges of Prof. Ian Stewart's novel Jack of All Trades.