Actually there has been a lot of fan controversey over the decades about how many times the Doctor has already regenerated.
Once the first Doctor regenerated into the second Doctor, it was possible to wonder whether he had ever regenerated before, and if so, how many times.
In episodes where different Doctors meet each other, they say which life or regeneration they are, and that agrees with their number as seen on the show, so it is logical to assume that the first, William Hartnell, Doctor was the first body the Doctor had, and that the regeneration into the second, Patrick Troughton, Doctor was the first regeneration.
In the 4th Doctor episode "The Deadly Assassin" it was said that a Time Lord only gets 12 regenerations and thus 13 bodies. So if the Doctor was only the 4th incarnation he would thus get 9 more regenerations and have 9 more bodies.
However, in the episode "The Brain of Morbius" there was a Time Lord duel between the Doctor and Morbius where each of them tried to force the other's mind back to before they were born, and thus to nonexistence. A screen showed the 4th Doctor's face, the 3rd Doctor's, the 2nd Doctor's, and the 1st Doctor's face. Then it showed eight more faces.
It was logical for fans to assume there had been eight previous incarnations of the Doctor before the first onscreen one, the William Hartnel Doctor, thus making the 4th onscreen Doctor, the Tom Baker Doctor, the 12th life. So when "The Deadly Assassin" was shown and mentioned the 12 regenerations rule fans worried the Doctor might have only one more regeneration left.
So fans usually decided that the eight faces seen before the William Hartnel Doctor were the faces of Morbius's 8 previous incarnations, and the official regeneration count of the Doctor supports this.
The number of previous incarnations of the Doctor was at first unclear within the series. In the Fourth Doctor story The Brain of Morbius (1976), the Doctor participates in a mental 'duel' with another Time Lord and the machine to which their minds are connected begins to project the faces of the "losing" contestant's regenerations in chronologically descending order. As the Doctor is overpowered by Morbius, the images change successively to those of the third, second and first Doctors, then eight further faces appear. It was the intention of producer Philip Hinchcliffe and script editor Robert Holmes that images were even earlier incarnations of the Doctor. However, the narrative does not explicitly confirm that the faces do represent incarnations of the Doctor, even though, as the faces are shown, Morbius cries "How far Doctor? How long have you lived?", indicating that they are not incarnations of Morbius, but rather that it is the Doctor's past being shown.
In other episodes, it was firmly established that the William Hartnell incarnation of the Doctor was the very first. In The Three Doctors, the Time Lord President describes the Hartnell incarnation as "the earliest Doctor". This is cemented in The Five Doctors when Peter Davison's Fifth Doctor (introducing himself to the First Doctor) says that he is the fourth regeneration, meaning that there have been five of him. The First Doctor also refers to himself as "the original". Episodes of the revival series showing the lives of the Doctor -- "The Next Doctor", "The Eleventh Hour", "The Day of the Doctor" and "The Husbands of River Song"—all begin with the William Hartnell incarnation. In "Twice Upon a Time", the William Hartnell incarnation is depicted as indeed being the very first incarnation with the First Doctor expressing fear about undergoing his very first regeneration throughout the story.
I have a theory that the Doctor is not a Galifreyan at all, but a being who dies and reincarnates into members of different species an endless number of times. So whenever he is reincarnated as a Galifreyan he gets 12 regenerations and 13 bodies within that one life as a Galifreyan.
Thus the faces from before the William Hartnell Doctor seen in "The Brain of Morbius" could be the faces of one or more previous lives as a Galifreyan or other human looking person. And when and if the last regeneration the Doctor dies, he will be reincarnated as another person, whether or not that is a Galifreyan.
And I haven't yet seen the last few seasons of the revived Dr. Who series and I don't know how they might affect the question of the number of regenerations of the Doctor.
Enclosed is a link to an article claiming the Doctor has had 32 Regenerations, which is partially based on evidence in an episode I haven't seen yet. I havent read the article to avoid spoilers when I do watch the later seasons.