The Minister

The Minister was a renegade Time Lord, and an associate of the Doctor going as far back as their third incarnation, when the Doctor helped cure the Minister's tenth incarnation of a condition which caused all their incarnations to have the same face, despite alterations in personality. At the time, the Minister had been living on the human colony planet Samael for nine consecutive incarnations, under the alias of Felicity Morgan.

The Minister's incarnation
In her original life on Gallifrey, the original incarnation of the Minister was genetically modified by the Celestial Intervention Agency so she would be able to retain her appearance across incarnations. Fleeing from Gallifrey in a TARDIS, she entered politics on the planet Samael, where the political system was defined by assassination. Under the name "Felicity Morgan", she was promoted nine times, being assassinated with each promotion and regenerating into the same face, her personality adjusting each time to fit her new position. Only her two closest assistants knew her secret.

In order, she became:
 * the Minister of Fisheries;
 * the Minister of Agriculture;
 * the Minister of Housing;
 * the Minister of Environment;
 * the Minister of Health;
 * the Minister of Education;
 * the Minister of War;
 * the Minister of Industry;
 * the Minister of Foreign Affairs and Diplomacy.
 * The latter's regeneration into the Prime Minister was witnessed by the Third Doctor and Jo Grant, who also helped her realise that she was being manipulated by the Master. With her consent, the Doctor removed the psychic block in her mind which kept all her regenerations the same. Despite knowing that her next regeneration would no longer have the face of "Felicity Morgan", she decided to keep going by "the Minister". (AUDIO: The Same Face)

Shortly after the Prime Minister's business with the Master and the Doctor was concluded, of her assistants joked that she might adopt the name of "Minister of Luck" in the future. (AUDIO: The Same Face) Notably, other accounts documented two incarnations of a renegade Time Lord going by "the Minister of Chance", an old friend of the Seventh Doctor. (WC: Death Comes to Time, etc.)