Bootstrap paradox

A bootstrap paradox was a sequence of events in which an event was among the causes of another event, which in turn is among the causes of the first-mentioned event. A common example was an individual who grew with a piece of music, then travelled back in time and introduced the song into the past, with the resultant question being: Who composed the song? (TV: Before the Flood)

Examples
Manning a Cyber-Scout craft, a Mondasian Cyber-Scout was sent on a mission to travel through a cosmic cloud that appeared the same time an ape-servant had arrived, to find the ape's home planet and convert their species. The Cyberman came through the cloud and was fired at by missiles by the Lizard Kings of Mondas, in an era before Mondas broke away from Earth. The missiles missed the Cyber-Scout craft, but the recoil injured it and damaged its voicebox. Noticing the Cyberman's technology was not unlike their own, the Lizard Kings decided to study the Cyber-Scout through vivisection and sent an ape-servant into the cloud to its world. The ape-servant inside the craft was the same ape that inspired the Cybermen to send the Cyber-Scout through the cosmic cloud in Mondas' future. (COMIC: The Prodigal Returns)

The Bad Wolf entity's existence was the result of a bootstrap paradox. Rose Tyler only looked into the heart of the TARDIS, creating the Bad Wolf, after she was motivated by seeing Bad Wolf graffiti, which she created after becoming the Bad Wolf herself. (TV: Rose, Aliens of London, The Parting of the Ways)

Donna Noble creates a bootstrap paradox by giving the idea of Miss Marple to Agatha Christie. Agatha got the idea of Miss Marple from Donna's knowledge of the future works by her. Although Agatha's memories of that night were wiped by her link with the Vespiform, the Tenth Doctor explains to Donna that some of the information may have "kept bleeding through... like Miss Marple". (TV: The Unicorn and the Wasp)

Joel Finch inadvertently inspired the pose of God in Michelangelo's Creation of Adam for the Sistine Chapel, after having been sent back to 1511 by a Weeping Angel. Joel studied the works of Michelangelo in the 21st century, and suggested this new pose based on his own recollections of the painting. Joel may have also invented the sandwich. (AUDIO: Fallen Angels)

It was the ArcHivist Hegelia's theory that the Isomorph Cybermen, the last remnant of the Neomorph Cyber-subspecies of 26th century Telos who were stranded in the 1980s after failing to change history by preventing the destruction of Mondas, made contact with the early CyberFaction on the dark side of the Moon. With the support and influence of the Isomorphs' cyber-technology, the CyberFaction would leave the Moon, splitting into separate subspecies that would ultimately come together and create the first of the Neomorph Cybermen on Telos. (AUDIO: The Ultimate Cybermen)

A bootstrap paradox was created while the Twelfth Doctor and Clara Oswald were at the Drum: the Doctor gained the idea of using a hologram of a ghost version of himself, through Clara who had seen the ghost Doctor and told the Doctor about it. (TV: Before the Flood)

For that matter, the circumstances that brought the Eleventh Doctor and Clara Oswald together came about largely because of a bootstrap paradox. The Doctor sought Clara out after having met two of her "splinters", at the Dalek Asylum, (TV: Asylum of the Daleks) and in Victorian London. (TV: The Snowmen) The Doctor eventually found the original Clara and took her on as a companion in the hopes of solving the mystery of how she seemed to have lived and died in different times and places. Their travels together would ultimately lead to the event that saw the creation of Clara's "splinters", including those that the Doctor previously met, thus setting him on course to find the original Clara to begin with. (TV: The Name of the Doctor)

According to one account, the Fifth Doctor's recognition of the phrase wibbly wobbly, timey wimey, (TV: Time Crash) seemingly coined by his tenth incarnation, (TV: Blink) was as a result of hearing River Song say it during an adventure she had with that incarnation. (AUDIO: A Requiem for the Doctor) River herself learned the phrase from the Eleventh Doctor, (TV: The Wedding of River Song, The Angels Take Manhattan) and assured the Fifth Doctor that it would grow on him in time. (AUDIO: A Requiem for the Doctor)

Amy Pond named her daughter, Melody Pond, after her childhood friend Mels, unaware that Mels was her daughter, making Melody her own namesake. Additionally, Melody started going by the name River Song after learning of her future identity from the Eleventh Doctor and her parents. (TV: Let's Kill Hitler)

Iris Wildthyme and Panda created a bootstrap paradox in order to divert a potentially dangerous future involving malevolent aliens from the future. (PROSE: Iris Wildthyme and the Unholy Ghost)

The involvement of the Dalek Restoration Empire in the Kotturuh crisis owed itself to a bootstrap paradox. (PROSE: The Last Message) After the Doctor had defeated the Dalek Time Squad, (PROSE: All Flesh is Grass, AUDIO: Mutually Assured Destruction) a surviving Dalek drone transmitted a message warning others about the Doctor's actions and the temporal fluctuations. Hearing this message, the Emperor of the Restoration decided to investigate, making sure to place the drone's past self on the Time Squad when he formed it. (PROSE: The Last Message)

When left Team TARDIS on a falling plane, the Thirteenth Doctor recorded a video to inform them how to survive, doing so because she had been informed by Ryan Sinclair that he had already done so. (TV: Spyfall)

When the Tenth Doctor found himself on Mira, he questioned who had deposited him on such a remote world, (AUDIO: Buying Time) later learning that he was the culprit, using a time tunnel to abduct his past self so he would be set on the path to stop the time tunnels. Privy to the Doctor thinking aloud, the Nun noted that she hated bootstrap paradoxes, a sentiment that the Doctor agreed with. (AUDIO: The Wrong Woman)

The creation of the Doctor Ogron was a bootstrap paradox. Upon first meeting his counterpart and being informed that the Ogron had arrived on Gallifrey in a future version of his own TARDIS, the Eighth Doctor agreed to travel to the Planet. While there, the Doctor's genetic material was used by the Overseer was used to create the Doctor Ogron. Understanding the paradox, the Doctor sent the Doctor Ogron to Gallifrey in his TARDIS to complete the loop. (AUDIO: Planet of the Ogrons)

Later in the Last Great Time War, the Eighth Doctor and Bliss travelled to parallel universe where they befriended an an alternate Davros. When the alternate Davros attempted to aid them in stopping the Dalek Strategist, he encountered a future version of himself, one corrupted by the Daleks, who manipulated into ensuring he would come to be, one who knew what to say because he remembered being told by his future self. (AUDIO: Palindrome)

When the Tenth Doctor and Martha Jones found themselves stranded in 1969, the Doctor used a transcript that he'd been given by Sally Sparrow in 2008 to communicate with her past self in 2007. The conversation that Sally had with the Doctor led to the completion of Larry Nightingale's transcript allowing Sally to gift it to a version of the Tenth Doctor who had yet to encounter her so he could use it in 1969. (TV: Blink)