SOURCE: The Mirror
Granting that The Mirror is not the most reputable source, the question raised by this article is a valid one. And the math - by the reckoning of most Whovians - does add up. If John Hurt's Doctor came between Paul McGann and Christopher Eccleston and the David Tennant Doctor did burn the energy for a regeneration during Journey's End then Matt Smith would indeed be The Doctor's Thirteenth incarnation.
So how can Peter Capaldi become The Doctor in The Time Of The Doctor then? Of course we knew they'd find some way around the rules regarding Time Lord regeneration... but how? With that in mind, here's five work-arounds I've come up with that would solve the problem.
1. The thirteen regeneration limit is a legal one.
While the mythology is firm on Time Lords being limited to a dozen regenerations, there is much to suggest the limit is a legal one controlled by The Time Lords rather than a physical one. The High Council of The Time Lords offered The Master a new set of regenerations for his aid during The Five Doctors. Further, The Trial of a Time Lord suggested it was legally possible for one Time Lord to sign their remaining regenerations over to another, as The Valeyard attempted to force The Sixth Doctor to do.
Therefore, with no Time Lords around to enforce their own limits, there's nothing to actually stop The Doctor from going on... assuming he found some way of absorbing the energy for a few more regenerations somewhere on his journeys. One way this could have happened...
2. River's sacrifice in Let's Kill Hitler didn't just save his life - it gave him more lives.
It was said that River sacrificed all her regenerations to save The Doctor from the poison that killed his ability to regenerate. What if she wound up doing a bit more than that?
3. As a result of The Night Of The Doctor, The Doctor's regeneration clock "reset".
In The Night Of The Doctor, we learn that the Paul McGann Doctor was left near-death following a spaceship crash onto the planet Karn - home of the Sisterhood of Karn, who protect the Sacred Flame that grants them (and The Time Lords) eternal life. They prepared several elixirs for him that would allow him to decide what he would become when he regenerated. Perhaps the elixir also gave him an entirely new set of regenerations?
4. The regeneration triggered by The Sisterhood of Karn was a 1-Up.
Another possibility is that the elixir The Sisterhood of Karn gave The Doctor functioned like an extra-life in a video game as well as a trigger for his next regeneration. That would give him one extra life, which could become the Peter Capaldi Doctor.
5. "Maybe even the universe can't bear to be without the Doctor."
In The Curse of The Fatal Death, The Doctor was somehow able to regenerate after his apparent death from an energy blast that destroyed his ability to regenerate. No explanation was given, save for the above phrase from his companion. Granting that The Curse of The Fatal Death was a spoof episode, it was also written by Steven Moffat and that scene was played completely straight.
Are any of these guesses close? Time will tell. It always does.