First Doctor's regeneration cycle

In accounts where the Doctor had Gallifreyan origins, they possessed a regeneration cycle of twelve regenerations and thirteen possible incarnations of which the First Doctor was the first incarnation. (TV: The Five Doctors, PROSE: Lungbarrow)

By some accounts, the Tenth Doctor's siphoned regeneration led to this cycle only having twelve incarnations, ending with the Eleventh Doctor, rather than reaching "thirteen silly Doctors". (TV: The Time of the Doctor, PROSE: Regeneration) Indeed, during the fall of Gallifrey, the Eleventh General believed that "all twelve" Doctors had appeared, only for Androgar to point out the thirteenth. (TV: The Day of the Doctor)

Izzy Sinclair, a companion of the Eighth Doctor, possessed a copy of a video game entitled Thirteen Doctors. (COMIC: Happy Deathday)

Origins
By most accounts, the First Doctor possessed his regeneration cycle from the moment of his creation. (PROSE: Lungbarrow, etc.)

By one account, the Doctor received his regeneration cycle when he got his Rassilon Imprimatur after graduating Prydon Academy. (PROSE: Divided Loyalties)

By accounts in which the Doctor was the Timeless Child, they originally possessed unlimited regeneration, with the limited cycle being an invention of Tecteun. Following their memories being wiped, the Doctor did not believe they could regenerate without limit. (TV: The Timeless Children)

First incarnation
The First Doctor was, by many accounts, the Doctor’s first incarnation. (TV: The Three Doctors, The Five Doctors, PROSE: Lungbarrow)

Second incarnation
Created by the First Doctor's regeneration, the Second Doctor was the Doctor's second incarnation. (TV: The Three Doctors, The Five Doctors, PROSE: Lungbarrow)

In one universe, another Doctor existed in the place of the Doctor's second incarnation. (AUDIO: Exile)

Third incarnation
During the Second Doctor's change of appearance, he was offered many options for his third incarnation but rejected them all, (TV: The War Games) eventually ending up with the Third Doctor as the third incarnation. (TV: The Three Doctors, PROSE: Lungbarrow, The Eight Doctors)

In the Inferno universe, the Second Doctor chose one of the faces offered to him, (PROSE: Timewyrm: Revelation) with his next incarnation becoming the Leader. He could no longer regenerate, as the Time Lords had taken this ability from him, and was ultimately killed. (PROSE: I, Alastair)

In another universe, the third Doctor was a woman. (AUDIO: Exile)

Another third Doctor existed in the Unbound Universe. (AUDIO: Sympathy for the Devil)

Fourth incarnation
Following the Third Doctor's regeneration, the Fourth Doctor was the Doctor's fourth incarnation. (TV: The Five Doctors, PROSE: Lungbarrow, The Eight Doctors)

Fifth incarnation
The Fourth Doctor's regeneration created the Fifth Doctor, the Doctor's fifth incarnation. (TV: The Five Doctors, PROSE: Lungbarrow, The Eight Doctors) Mawdryn attempted to force the Doctor to use up his eight remaining regenerations to end his follower's cycle of perpetual rebirth, but this was rendered unnecessary when Alistair Gordon Lethbridge-Stewart made physical contact with his younger self and a discharge of temporal energy was released that allowed Mawdryn and his followers to die. (TV: Mawdryn Undead)

In one reality, the Fourth Doctor was forced to regenerate prematurely during the Genesis Incident, becoming the Warrior. (AUDIO: Dust Devil)

Sixth incarnation
The Fifth Doctor's regeneration created the Sixth Doctor, the Doctor's sixth incarnation. (PROSE: Lungbarrow, The Eight Doctors) After the TARDIS became "stalled in the equivalent of a galactic lay-by", the Doctor had a worried thought of Peri Brown growing old and dying in the TARDIS, while he would "go on regenerating until all [his] lives [were] spent". (TV: Vengeance on Varos)

When exposed the Valeyard's alliance with High Council to the Sixth Doctor at his trial, he revealed that the Valeyard was acting as the prosecutor for the trial in exchange for the Doctor's remaining regenerations. (TV: The Ultimate Foe)

Another sixth incarnation was briefly created after a premature regeneration of the Fifth Doctor. (AUDIO: The People Made of Smoke)

Seventh incarnation
The Sixth Doctor's regeneration created the Seventh Doctor, the Doctor's seventh incarnation. (PROSE: Lungbarrow, The Eight Doctors)

The Sixth Doctor was brought to the brink of a regenerative collapse when he was gravely wounded by Zor before being saved by Captain Jack Harkness. Whilst the Doctor was incapacitated, Jack took the Doctor's costume and posed as him, claiming to have regenerated. (AUDIO: Piece of Mind)

Eighth incarnation
The Seventh Doctor's regeneration created the Eighth Doctor, the Doctor's eighth incarnation. (PROSE: The Eight Doctors) In an alternate timeline where the Seventh Doctor regenerated prematurely, the Eighth Doctor was known as Johann Schmidt. (AUDIO: Klein's Story)

After using a Deathworm Morphant to possess a human body, tried to use the Eye of Harmony to steal the Eighth Doctor's remaining regenerations to heal himself, but his plans were foiled when Grace Holloway sent the TARDIS' into a temporal orbit. (TV: Doctor Who)

When talking about the future, the Eighth Doctor noted he would die again, and that one day he would "run out of deaths". (AUDIO: The Gift)

Gloria Swannicker once faked the Seventh Doctor's death and pretended to be his successor. (AUDIO: A Life of Crime)

Performing Eighth Man Bound, the First Doctor once saw all of his potential future incarnations up to the Eighth Doctor. (PROSE: Christmas on a Rational Planet)

Ace once saw the eighth incarnation in the Doctor's mind in its preformed state as a Watcher, a blurred pale man not yet of fixed face. (PROSE:  Revelation) Bernice Summerfield once saw the Eighth Doctor in the Seventh Doctor's mind, although the Seventh Doctor believed that he was only a possible eighth incarnation. (AUDIO: The Shadow of the Scourge)

When the Seventh Doctor once momentarily regenerated, he momentarily had the face of another man. (PROSE: Parasite)

Ninth incarnation
The Eighth Doctor's regeneration led to the Doctor's ninth incarnation, although the complex temporal lifetime of the Eighth Doctor led to their being multiple candidates for this incarnation. (PROSE: The Gallifrey Chronicles)

In a possible future for the Eighth Doctor, (PROSE: The Tomorrow Windows) the "listless-looking" Doctor who traveled with Emma was, as the Master knew, the "ninth body" with "many, many more" to come. (TV: The Curse of Fatal Death)

In another possible future, (PROSE: The Tomorrow Windows) the Doctor who travelled with an android Master once mused that, like a cat, he had used up his nine lives. (WC: Scream of the Shalka)

The Ninth Doctor was, by some accounts, the Doctor's ninth incarnation. (COMIC: The Forgotten)

Other accounts indicated the War Doctor was the Doctor's ninth incarnation. (TV: The Time of the Doctor, PROSE: Regeneration)

Ace once saw the ninth incarnation in the Doctor's mind in its preformed state as a Watcher, a blurred pale man not yet of fixed face. (PROSE:  Revelation)

Tenth incarnation
The Tenth Doctor, created by the Ninth Doctor's regeneration, was by some accounts the Doctor's tenth incarnation. (COMIC: The Forgotten, The Age of Ice)

In the future in which they traveled with Emma, the "listless-looking" Ninth Doctor regenerated into the Tenth Doctor. (TV: The Curse of Fatal Death)

By other accounts, created by the War Doctor's regeneration, the Ninth Doctor was the Doctor's tenth incarnation. (PROSE: Regeneration, TV: The Time of the Doctor)

Greeneye once pretended to be the Doctor's tenth incarnation. (PROSE: Human Nature)

Ace once saw the tenth incarnation in the Doctor's mind in its preformed state as a Watcher, a blurred pale man not yet of fixed face. (PROSE:  Revelation)

Eleventh incarnation
The Eleventh Doctor, created by the Tenth Doctor's regeneration, was shown to believe that he could still regenerate. (TV: Let's Kill Hitler) His successor the Twelfth Doctor, by some accounts, was the Doctor's twelfth incarnation. (PROSE: Twice Upon a Time, COMIC: Four Doctors) Clara Oswald believed he was "number eleven", although he would dispute this. (TV: The Time of the Doctor)

By other accounts, created by the Ninth Doctor's regeneration, the Tenth Doctor was the Doctor's eleventh incarnation. (PROSE: Regeneration) The Tenth Doctor's siphoned regeneration was the Doctor's eleventh regeneration, meaning the Tenth Doctor's vanity prevented the regeneration cycle from having thirteen incarnations. (TV: The Time of the Doctor) This siphoned regeneration transformed the Tenth Doctor's severed hand and Donna Noble into the Meta-Crisis Doctor and the DoctorDonna respectively. (TV: Journey's End)

In the future in which they traveled with Emma, the Tenth Doctor regenerated into the Eleventh Doctor. (TV: The Curse of Fatal Death)

Ace once saw the eleventh incarnation in the Doctor's mind in its preformed state as a Watcher, a blurred pale man not yet of fixed face. (PROSE:  Revelation)

Twelfth incarnation
Created by the Tenth Doctor's regeneration, the Eleventh Doctor was identified as the twelfth incarnation (TV: The Time of the Doctor, PROSE: Regeneration) and the final incarnation of the regeneration cycle which began with the First Doctor. (TV: The Time of the Doctor, COMIC: Four Doctors) Indeed, Clara Oswald said that he was "number twelve". He was fated to be the Doctor's final incarnation, but was saved from this by the Time Lords granting the Doctor's new regeneration cycle. (TV: The Time of the Doctor)

The Tremas Master once told the Sixth Doctor that the Valeyard was "an amalgamation of the darker sides of your nature, somewhere between your twelfth and final incarnation". (TV: The Ultimate Foe) The Valeyard once told the Ashmael about the role of the Doctor’s twelfth incarnation in the Valeyard's creation. Ashmael told the Sixth Doctor that it was fascinating, and asked the Sixth Doctor if he had any idea what his twelfth incarnation would be like. (PROSE: Millennial Rites)

In the future in which they traveled with Emma, the Eleventh Doctor regenerated into the Twelfth Doctor. (TV: The Curse of Fatal Death)

Ace once saw the twelfth incarnation in the Doctor's mind in its preformed state as a Watcher, a blurred pale man not yet of fixed face. (PROSE:  Revelation)

Thirteenth incarnation
By some accounts, this cycle never reached its thirteenth incarnation due to the Tenth Doctor's siphoned regeneration, meaning that while the Twelfth Doctor was the Doctor’s thirteenth incarnation, he was of another cycle. (PROSE: Regeneration, TV: The Time of the Doctor)

By several accounts, the Doctor's thirteenth incarnation was the Doctor's final incarnation. (PROSE: Millennial Rites, The Ultimate Foe)

In the future in which they traveled with Emma, the Twelfth Doctor regenerated into the Thirteenth Doctor after being exposed to a zectronic beam, much to the astonishment of the Master. (TV: The Curse of Fatal Death)

Ace once saw the thirteenth incarnation in the Doctor's mind in its preformed state as a Watcher, a blurred pale man not yet of fixed face. (PROSE:  Revelation)

Incarnations of ambiguous numbering
One bearded Doctor was grouped along with the Doctor's first seven incarnations in UNIT files. (PROSE: An Army of Shadows)

Beyond the regeneration cycle
The Vortex Butterfly once told the Tenth Doctor that, in the Ocean of Time, he had far more than just thirteen lives. In truth, he extended both forwards and backwards, infinitely and looping. (COMIC: Vortex Butterflies)

According to the mindbending duel with Morbius, there had been eight earlier faces of the Doctor before the First Doctor, a sequence in which the Fourth Doctor numbered twelfth. (TV: The Brain of Morbius)

The Fourth Doctor told Romana II that Time Lords had ninety lives, and that he had already gone through "about a hundred and thirty" of them. (TV: The Creature from the Pit) The Fifth Doctor would remember this statement as being connected to the faces seen in the mind duel with Morbius. (PROSE: Cold Fusion)

Behind the scenes

 * Following the introduction of the War Doctor and the dialogue of The Time of the Doctor, the incarnation numbering of this life cycle was thrown into question, with the Eleventh Doctor as the twelfth incarnation, something repeated in sources such as Doctor Who: The Essential Guide. The Time Lord Letters would refer to the Twelfth Doctor as the Doctor's twelfth or thirteenth incarnation, depending on accounts. However, Terror Moon has the Twelfth Doctor referring to the Tenth Doctor as his "tenth incarnation". Later sources such as DWM 584 would start capitalizing the numbers of incarnations, referring to the Tenth Doctor as the Doctor's Tenth incarnation.
 * A cut scene from Return of the Cybermen would have had the Time Lord messenger inform the Fourth Doctor that fallout from the coming Time War would result in his "seventh and eleventh bodies" "living out the same events twice", referring to the novel Human Nature and the televised adaptation Human Nature featuring the Seventh and Tenth Doctor respectively.

Notvalid incarnations of this regeneration cycle

 * Trevor Martin's incarnation of the Doctor was shown to be a successor to the Third Doctor in The Seven Keys to Doomsday.
 * Lenny Henry's incarnation of the Doctor was shown to be a successor to the Sixth Doctor in The Lenny Henry Show.

Notvalid imposters

 * Captain Jack Harkness (John Barrowman) found Sao Til (Tim Ingham) in the TARDIS, who claimed to be the Doctor having regenerated. Captain Jack understood otherwise and confronted him, only for the pair to be interrupted by Tenth Doctor actor David Tennant. (NOTVALID: Tonight's the Night)