Unfortunately Oni, if we're including the war doctor, we are on the 13th doc.
The War Doctor was not a 'natural' occuring regeneration for a time lord. It was forced on him
By the sisterhood of Carn, which by the way is something that was started after the Time Lords were 'cursed' (not all apparently) with the inability to genetically reproduce without 'Looms'.
. Also the whole 'only this many regenerations' was NOT something that was a hard line. The fact of the matter was the Time Lord Council was full of a buncha pompous fuckwits that wanted to control EVERY aspect of TIme Lord Society. This included how many regens they got. There is no actual limit to the Regens the Counsil says
there is only that many regenerations, when in reality there isn't. THe Master has gone above and beyond his regeneration limit, and so has many
other time Lords historically. The 'limit' was put into place because the TIme Lord Council highly monitored their population, they figured 'well you used that many your time is done, you don't get to regenerate again' - because they had to Loom children after the curse placed upon them. Looming involves a genetic process, but it isn't like repopulating through sex. Instead only a set number of children was allowed to join each house, as another member left. None of the Time Lords saw this as an issue, not at first, because 'Duty and Tradition' told them it was the right
thing to do.