Regeneration cycle

The regeneration cycle, (TV: The Deadly Assassin, The Time of the Doctor, et al.) also called the life cycle (TV: The Five Doctors, PROSE: First Frontier) or regenerative cycle, (PROSE: No Future, Lungbarrow, Dalek: The Astounding Untold History of the Greatest Enemies of the Universe, AUDIO: Zagreus) was the set of twelve regenerations and thirteen incarnations which made up a Time Lord's lifespan. (TV: The Deadly Assassin, et al.)

At the end of their lifecycle, Time Lords faced permanent death. (TV: The Five Doctors) As such, while the Time Lords possessed the ability to "cheat death", as the Ninth Doctor put it, (TV: The Parting of the Ways) they were not truly immortal, which the Fifth Doctor believed impossible even for them. (TV: The Five Doctors) In certain cases, however, the Time Lords proved able to bestow additional regenerations. (TV: The Five Doctors, The Time of the Doctor)

Regeneration limit
The regeneration cycle was composed of twelve regenerations, which resulted in thirteen incarnations. (TV: The Deadly Assassin, et al.)

The Eleventh Doctor once claimed to Clyde Langer that he could regenerate 507 times. (TV: Death of the Doctor)

Premature end
As noted by the Tenth Doctor, a Time Lord could die prematurely should they be killed before regeneration could occur. (TV: The End of Time) As well as that, weaponised inhibitors were developed with the purpose of preventing regeneration. (AUDIO: The Conscript)

After Missy was fatally wounded by the laser screwdriver beyond the ability to regenerate, (TV: The Doctor Falls) she required a new regeneration cycle to regenerate. (AUDIO: The Lumiat)

End of the cycle
Without access to additional regeneration energy, a Time Lord would ultimately perish at the end of their thirteenth incarnation. (TV: The Time of the Doctor)

In his thirteenth and final body, Azmael deliberately regenerated past his limit, killing him and Mestor, who had been attempting to possess Azmael's body after his own was destroyed. (TV: The Twin Dilemma)

As witnessed by the Sixth Doctor and Peri Brown, an elderly Time Lord at the end of his final life disintegrated as he "degenerated" into molecules. (COMIC: The World Shapers)

During the Last Great Time War, Vassarian began to regenerate as an Eternity Cage sealed around him. With the process unable to complete, Vassarian burnt through all his regeneration energy and lives. (AUDIO: The Eternity Cage)

Origins
It was believed that a Time Lord could only regenerate twelve times because their symbiotic nuclei could only divide so much. The Valeyard, however, claimed this was imposed by Rassilon to prevent a thirteenth regeneration and keep Time Lords from immortality. (AUDIO: Trial of the Valeyard)

The Doctor was revealed to be the progenitor of the Time Lords, being the Timeless Child, they had unlimited regenerations. After the Time Lords redacted their memory, (TV: The Timeless Child) the Doctor appeared to be limited to twelve regenerations, until they were seemingly given a new regeneration cycle by the Time Lords. (TV: The Time of the Doctor)

Granting of additional regeneration cycles
By some accounts, Time Lord officials were able to grant Time Lords additional complete regeneration cycles after their original cycles were exhausted. (TV: The Five Doctors, et al.)

Nature
Perpetual regeneration.

By another account, Time Lords gained new regenerative cycles by loom jumping. (PROSE: Lungbarrow)

Origins
Concerning the process' origins, the author of A Brief History of Time Lords once stated, "We’ve long (always?) been able to dole out complete new regeneration cycles." (PROSE: A Brief History of Time Lords)

During the Last Great Time War, the Time Lords acquired the Matrix print of Artron, a Gallifreyan scientist from Rassilon's era who had been experimenting on means of extending the natural ability to regenerate. Thanks to his research, Artron was able to devise a means of not only granting Time Lords new regenerations, but even grant non-Time Lords potentially unlimited regenerations without actually changing in appearance or personality. When this print was acquired by the War Master, he returned it to the other Time Lords after working with a past and future version of himself to restore another past Master to life as part of his deal with the Celestial Intervention Agency, as they thought he would make a perfect warrior. (TV: The Sound of Drums, AUDIO: The Devil You Know, Assassins). This restoration granted the younger Master a complete new regeneration cycle, although the Master was explicitly forbidden to give himself anything more than a standard set of regenerations. (AUDIO: Day of the Master)

Uses
Despite the limit of thirteen lives, the High Council of Gallifrey possessed the ability to bestow additional regenerations upon individual Time Lords. (PROSE: A Brief History of Time Lords) While was offered a complete life cycle by Borusa, (TV: The Five Doctors) the Eleventh Doctor was granted an indeterminate amount of additional regenerations through regeneration energy provided from Gallifrey. (TV: The Time of the Doctor, Kill the Moon, Hell Bent) Mantus tempted Narvin with the promise of a new regeneration cycle to ensure loyalty to Rassilon, as Narvin could no longer regenerate, (AUDIO: Assassins) having had his remaining lives stolen from him on an alternative timeline's Gallifrey. (AUDIO: Reborn) This process was incredibly painful, and was halted at Narvin's request before reaching completion. (AUDIO: Assassins)

Without this option, certain Time Lords would seek to extend their lives through other means. Having used all his lives, the Master took to possessing the bodies of non-Gallifreyans. (TV: The Keeper of Traken, Doctor Who, AUDIO: Mastermind) The Fifth Doctor knew offhand that Numismaton Gas, a gas with enormous "transformative powers", could be "a great help" to a Time Lord who had run out of regenerations. Indeed, after the Master exposed himself to Numismaton fire, (TV: Planet of Fire) the Celestial Intervention Agency came to believe (PROSE: CIA File Extracts) erroneously (PROSE: First Frontier, TV: Doctor Who, etc.) that the Master had gained a new set of regenerations from the experience. (PROSE: CIA File Extracts)

In theory, Time Lords could also be robbed of their regenerations by others of their kind, as the Master attempted to do to the Eighth Doctor via the Eye of Harmony. (TV: Doctor Who) The Master also believed it was possible to acquire new regenerations directly from the Eye itself, though only at the cost of releasing a force that would destroy Gallifrey. (TV: The Deadly Assassin)

In a story relayed by the Valeyard, the thirteenth and final incarnation of the Doctor supposedly composed a set of scrolls which detailed his work to bypass the limit of twelve regenerations. The Valeyard's claim, however, was doubted by the Sixth Doctor. Indeed, the Sixth Doctor found what appeared to be his final incarnation on Etarho to have been the Valeyard masquerading as him. (AUDIO: Trial of the Valeyard)

An Elysian field was a method, using cryogenetic extraction, that allowed a Time Lord to completely break down their body into its components, i.e. atoms and molecules, and rebuild themselves, editing their personality. It could spark a regeneration, even at the end of a regeneration cycle, and could be used to imbue its user with a new regeneration cycle. Because of its ability to enable its user to cheat death, and because it left them "unbound" from their previous lifetime (able to interact with, or even destroy, their earlier self with impunity without fear of causing a paradox), the technology was banned by the Time Lords. Notably, after being shot by her in a way that inhibited her ability to regenerate,  used an Elysian field to regenerate into a incarnation which called herself the Lumiat. This also imbued the Master with an entirely new regeneration cycle. (AUDIO: The Lumiat)

The Master
The Master had expended his initial life cycle, but cheated death by possessing the body of the Trakenite Tremas. (TV: The Keeper of Traken) In return for retrieving the Doctor from the Death Zone on Gallifrey, Lord President Borusa offered to bestow upon the Master a new life cycle. Ultimately, the Master did not receive the life cycle from the High Council. (TV: The Five Doctors)

Eventually, however, the Master, having been resurrected by the Time Lords to fight in the Last Great Time War, possessed the ability to regenerate again. (TV: Utopia, The Sound of Drums, AUDIO: Eyes of the Master, The Devil You Know) This gave the Master a new set of twelve regenerations. (AUDIO: Day of the Master, TV: Utopia, Dark Water) When Missy chose to stand with the Twelfth Doctor, the furious Saxon Master fatally wounded her beyond the ability to regenerate with his laser screwdriver. (TV: The Doctor Falls) Using an elysian field, Missy managed to regenerate, granting herself a new cycle of twelve regenerations. (AUDIO: The Lumiat, TV: Spyfall)

The Doctor
Another Time Lord who lived beyond their initial life cycle was the Doctor. (TV: The Time of the Doctor) However, there was confusion as to how many regenerations he used in his first cycle. Having abandoned their title for the vast majority of their actual ninth incarnation, (TV: The Day of the Doctor) and wasting a regeneration to prevent changing their current form after their eleventh regeneration, the so-called Eleventh Doctor confirmed he had used all of his regenerations and was in his final incarnation. On the verge of dying of old age during the siege of Trenzalore, the Doctor's companion Clara Oswald pleaded with the Time Lords through a crack in the universe to change the future and save the Doctor. In response, the Time Lords gave the Doctor a new life cycle by sending him regeneration energy through the crack. This caused an explosive "reset" which restored the Doctor to his youthful form and destroyed the attacking Dalek forces but still required the Doctor to regenerate into his next incarnation. (TV: The Time of the Doctor)

However, the Twelfth Doctor remained uncertain of the number of regenerations he possessed, (TV: Kill the Moon) with even Rassilon not being sure how many were granted to him (TV: Hell Bent) and expressing uncertainty about the Doctor's new limits as well. (TV: The Doctor Falls)

The Valeyard, having "splintered" from the Doctor as a causal imbalance, was limited to only one life and could not regenerate. (AUDIO: Every Dark Thought)

The Thirteenth Doctor encountered an incarnation of herself that she believed was from her own past though the Doctors possessed no memory of each other. (TV: Fugitive of the Judoon) Not long afterwards, the Spy Master claimed to the Thirteenth Doctor that she was originally the Timeless Child — before the first life she remembered — who was able to regenerate an unknown number of times, from whom Time Lord regeneration was derived. In this account, the life cycle was artificially after the ability to regenerate was genetically engineered into the Time Lords who lived in the Capitol. (TV: The Timeless Children)

The Spy Master later forced the Thirteenth Doctor to regenerate into himself. However, the Doctor's companions were able to trigger a retro-regeneration using all of the regeneration energy of a group of CyberMasters, forcing the Master back into his own body and transforming the Thirteenth Doctor back to normal.

After being mortally wounded by the Spy Master, the Thirteenth Doctor underwent a regeneration into an incarnation identical to the Tenth Doctor, albeit with a slightly older appearance. (TV: The Power of the Doctor)

Proto-Time Lords
While possessing the ability to regenerate, the life cycles of the Proto-Time Lords created by the Kovarian Chapter were uncertain, with separate individuals possessing different numbers of regenerations. (AUDIO: The Lady in the Lake) The original, River Song, regenerated twice before sacrificing the remainder of her regeneration energy to save the Eleventh Doctor from death. (TV: Let's Kill Hitler)

Lake regenerated twice. Following his first regeneration, Lake, unaware of the limit of his regenerative ability, proceeded to test it by inflicting fatal injuries against his fellow Proto-Time Lords while convincing others to commit suicide. To that end, he was responsible for the deaths of Rindle, Wadi, Creek, Beck, Stream, all of whom were determined to possess a different number of regenerations. Rindle regenerated four times before their remaining regenerations were expended by Lake to power his ship. Ironically, Lake's last victim was his third and final incarnation, a girl who became known as Lily having lost her memory of her prior lives. Having witnessed Lake regenerate into Lily after mortally wounding him in anger for murdering her, River Song was forced to send Lily back in time so she would meet her fate. Tarn had regenerated at least ten times. (AUDIO: The Lady in the Lake)

Brooke, one of the second batch of Proto-Time Lords, was observed to regenerate at least once. (AUDIO: My Dinner with Andrew)

Other
The Tenth Doctor suggested that Jenny, who was artificially created from his DNA and so had a binary vascular system, was capable of regeneration only for her to fail to do so when she was fatally shot, with Martha Jones suggesting that she was "like [the Doctor], but maybe not enough." However, Jenny eventually came back to life. (TV: The Doctor's Daughter)

The Meta-Crisis Tenth Doctor, who was created from a meta-crisis between the Tenth Doctor's hand and Donna Noble, was physically human to the point that he could not regenerate. (TV: Journey's End)

Behind the scenes
The regenerative limit of thirteen lives was first established in the television story The Deadly Assassin, which featured 's attempt to cheat his final death.