Regeneration cycle

A regeneration cycle or life cycle referred to the number of regenerations a Time Lord normally possessed. Time Lords normally had twelve regenerations, which resulted in thirteen incarnations before 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 Sound of Drums, The Time of the Doctor)

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)

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)

Curiously, the Eleventh Doctor, who had already expended all twelve of his regenerations, (TV: The Time of the Doctor) once claimed to Clyde Langer that he could regenerate 507 times. (TV: Death of the Doctor) However, he was being flippant and admitted to Clara Oswald that he was on his final life. (TV: The Time of the Doctor)

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).

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)

Beyond thirteen lives
Despite the limit of thirteen lives, the High Council of Gallifrey possessed the ability to bestow additional regenerations upon individual 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)

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) 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)

In a story related 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 Doctor found what appeared to be his final incarnation on Etarho to have been the Valeyard masquerading as him. (AUDIO: Trial of the Valeyard)

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 Gallifrey. (AUDIO: Reborn) This process was incredibly painful, and was halted at Narvin's request before reaching completion. (AUDIO: Assassins)

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, 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) Like the Doctor, it is unclear how many regenerations the Master was given in his new cycle, although his was ordered by the Celestial Intervention Agency to only give himself a standard set of thirteen lives. After being destroyed by his past and future selves, the Master regenerated three times and expected to be able to regenerate at least once more. (TV: Utopia, Dark Water, The Doctor Falls, Spyfall; COMIC: Fast Asleep, AUDIO: Day of the Master)

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. It is unclear how this incarnation could exist given the limits of the life cycle. (TV: Fugitive of the Judoon)

The Thirteenth Doctor later discovered 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 limit was imposed only afterwards. (TV: The Timeless Children)

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 artifically 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.