You want some overthinking?
Ok here goes. The whole 13 regeneration cycle doesn't apply to The Doctor, because he is a mutant aberration.
Here's my reasoning. Who else do we know changes his face mysteriously yet carries the same name? He's british and always fixes the problem before him.
That's right it's ever ones favourite spy James Bond. So here's my theory.
Every time the Doctor regenerates, he gets thrown out of the time vortex (that's what the flashy orange stuff is REALLY all about) where mid-regeneration he is captured by MI6 and memory partitioned as he reforms. His mind then goes from track A, helpful time travelling timelord, to track B murdering git with a fancy car. When James Bond dies his inevitable untimely death at the hands of a female assassin, the re-generation triggeres another leap through the vortex back to the Tardis, then to his last point of death, where he becomes a new regeneration all together.
So now we know why two famous British characters both change faces, one without explanation.
Or perhaps he does know about it, for we've seen that a regeneration was discounted because he became a warrior (or spy). Perhaps that regeneration saw his track A and B's working together.
There you go, some epic overthought for you, that never-the-less explains so much.
Anyway the point is, that he's obviously had WAY more than 13 regenerations, which makes him a mutant timelord with MI6 assassination skils.
Btw this is from wikipedia which I found interesting.
Quote: ""The number of previous incarnations of the Doctor was at first unclear within the series. In the Fourth Doctor story The Brain of Morbius, 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,[15][16] although the narrative does not definitively assert that these are past incarnations of the Doctor rather than of Morbius.
In later episodes, it was firmly established that the William Hartnell incarnation of the Doctor was the very first. For example, it is explicitly stated by the Fifth Doctor in Mawdryn Undead that he has eight incarnations left. This is further stated in The Five Doctors when Peter Davison's Doctor (introducing himself to the First Doctor) says that he is the fourth regeneration, meaning that there have been five of him. Also in this serial, the First Doctor refers to himself as the "original". In the TV movie, the Doctor (played by Paul McGann) say that he was "nearing the end of [his] seventh life" when referring to the Sylvester McCoy incarnation in flashback. The Doctor also states that "A Time Lord has thirteen lives and The Master had used all of his".""
So the reality is, that we don't even know the doctors real name lol, we when the first series was done he was already an old man, he might have regenerations before that. The ultimate reason for all this however is the following.
He has extra regenerations now because doctor who is doing quite well world wide, has the title of the oldest and longest running scifi series ever, and so will probably make money for the next 50 years, plus there's a little bit of national british pride involved. Because of all this he gets extra regenerations because...
it was written into the script
