Renegade Time Lord (The Eleven)

A Renegade Time Lord who suffered from regenerative dissonance faced at least the Fourth, Fifth, Sixth, Seventh and Eighth Doctors.

Name
Each incarnation of this Time Lord went by a sequential number, for example the eleventh incarnation was known as "the Eleven". (AUDIO: The Eleven, et al)

When the Nine sought to make himself immortal using the Crucible of Souls, he considered several new names, such as "the Infinite", "the Numberless", and "the Eternity", before concluding that naming things was much more complicated than it was made out to be. (AUDIO: The Crucible of Souls)

Incarnations
In their first incarnation, this Time Lord was a member of the High Council and worked in the Panopticon Archive. (AUDIO: The Eleven, The Satanic Mill)

The Two was described by the Eighth Doctor as "the charmer". (AUDIO: Planet of the Ogrons)

According to the Eighth Doctor, the Three was one of the personalities more willing to kill. (AUDIO: The Eleven)

The Four was described by Liv Chenka as "the arrogant one". (AUDIO: Day of the Master)

The Five boasted that he was a formidable opponent who was capable of killing anyone. (AUDIO: Stop the Clock)

The Six was by far the most violent of the Time Lord's incarnations, and had one of the most unstable personalities. He would often try to get the Eleven to butcher his way through a situation, regardless of whether the victims could be of use. (AUDIO: The Eleven)

The Seven was known for being rude and abrasive. Once, he was used by the Eleven to portray a scientific advisor. (AUDIO: World of Damnation, Sweet Salvation)

As the Eighth Doctor described, the Eight was "the exception that proved the rule". (AUDIO: Planet of the Ogrons) Through meditation, he was able to suppress the voices of his previous incarnations. While assisting the Doctor and "Sister Cantica" in defeating the Clocksmith, he was shot. After retreating to the Doctor's TARDIS, he regenerated. (AUDIO: The Eighth Piece / The Doomsday Chronometer)

The Nine initially claimed to be a new incarnation of the Doctor to Helen Sinclair and Liv Chenka. Unlike his predecessor, the Nine did not try to be a "good man" and was motivated by kleptomaniacal greed. (AUDIO: The Crucible of Souls, Companion Piece, Relative Time)

The Ten was known for his skills in hypnotism. (AUDIO: Stop the Clock)

The Eleven was captured by the Seventh Doctor and imprisoned on Gallifrey. (AUDIO: Dark Universe) However, he managed to escape, and so the Eighth Doctor was brought in to find him. (AUDIO: The Eleven) Along with Padrac, Caleera and the Clocksmith, the Eleven was a member of the Doom Coalition, a secret organisation of Time Lords who sought to bring about the end of the universe. (AUDIO: The Crucible of Souls, Stop the Clock)

While running from the Ravenous, the Eleven met his ninth incarnation, who was attempting to get revenge on the Doctor, and learnt that his regenerative dissonance made him immune to being consumed by the Ravenous. (AUDIO: The Odds Against) After entering an alliance with the Ravenous, the Eleven was foiled by the Doctor, Liv Chenka, Helen Sinclair and three different incarnations of the Master. (AUDIO: Planet of Dust) On the Crucible of Souls, he met his comeuppance and was killed by the three Masters while trying to form an alliance. Left for dead, the Eleven regenerated into the Twelve, a female incarnation. (AUDIO: Day of the Master)

During the Last Great Time War, the Twelve was able to stabilise her regenerative dissonance via a neural inhibitor implanted into her brain by the Time Lords. After reuniting with the Eighth Doctor, the Twelve travelled with him and Bliss to the Ogron homeworld and infiltrated a Dalek base commanded by the Overseer, (AUDIO: Planet of the Ogrons) before the Daleks imprisoned her with the Doctor and Bliss on Sangrey for interrogation, disabling her neural inhibitor in the process. (AUDIO: In The Garden of Death) After a mission for the Time Lords on Uzmal, during which the Daleks used a mind probe to control her, the Twelve was imprisoned in cryo-freeze by the Time Lords on Gallifrey, (AUDIO: Jonah) but she eventually escaped after the Daleks were erased from history. The Twelve then utilised a Dreadshade and attempted to blackmail the High Council into granting her a seat on the Council, but ultimately failed, and after causing the Dreadshade to panic and fire a Bolt of Frightening, the Twelve was seemingly buried in a large amount of rubble. (AUDIO: Dreadshade)

Following the end of the Time War, the Ninth Doctor believed all the Time Lords bar himself to be dead, (TV: Dalek) and so told Liv that he did not expect to see the Time Lord once known as the Eleven again. (AUDIO: Flatpack)

Behind the scenes

 * , no name has been given in-narrative for this Time Lord in general (only for each of their incarnations). However, John Dorney, who regularly wrote stories featuring the character, has expressed a fondness for the name "the Collective".
 * refers to them as "that numerical non-entity" in 2019's Day of the Master.