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The Doctor of Doctor Who is effectively immortal. On those rare occasions when he dies, he regenerates into a new form, often with a new personality. It's a fine time to switch actors, which is probably good because William Hartnell (who portrayed the First Doctor) would be 103 if he still lived.
Brian Rimmer, who once assembled a fine video showing the introductory theme to every Doctor Who series, has compiled the regeneration scenes of all eleven Doctors. The last one at 8:45 is just plain funny.
For a start, although it is now generally accepted that when Hartnell changed to Troughton it was a regeneration, it has been stated in some official sources that this was not a regeneration, but a rejuventation. So there's one Doctor they can add to the apparent limit of 13.
From Troughton to Pertwee the timelords forced a "change of appearance" upon the Doctor. It wasn't referred to as a regeneration at the time, so the writers can use that to increase the number of Doctors to 15 should they so choose.
And of course we also know the Master ran out of regenerations and simply took over another body. So that would mean that in theory the Doctor could go on living forever simply by taking over a new body. And he was also offered a complete new cycle by the High Council. So the limit of 12 regenerations would not appear to be a natural one, but something imposed by the High Council. If you were a writer in need of a get out you could assume that there being no High Council there is no longer a limit to a time lord's regenerations.
We all know that the Doctor lies (rule one), but he claimed recently that he can regenerate 507 times. The Doctor is far from consistent in this respect. In the Five Doctors Hartnell's Doctor states that he is the original, but in the fourth Doctor adventure The Brain of Morbius during contest of the mind the Doctor generates images of what are apparently his previous incarnations. We see the three faces we are expecting and many more besides, suggesting that Hartnell's Doctor was perhaps not the original after all. Like many aspects of the lore of time lords this is never explained in the series.
When the fifth Doctor is almost convinced to give up his remaining regenerations to save his companions he says he would cease to be a time lord as a result. Before River Song gives up her remaining regenerations to save the Doctor it is stated that the Doctor's regenerative cycle was suspended. It is not explained whether, as a result of River's intervention, the Doctor's regenerative cycle has been restored and whether or not the Doctor has reveived River's ten remaining lives. Always assuming, of course, that River has the standard (if so it is) 13 lives. Only being part timelord she may have a different number of regenerations. As such we are now totally in the dark as to how many regenerations the Doctor has left - even if we accept that before Let's Kill Hitler he had two left (and I don't).
As an aside don't assume that "the fall of the eleventh" refers to the eleventh doctor. Nobody has clearly stated that it does, it could be many things. How about the fall of the Eleventh Cyber Legion, for example. Assume nothing.
Go back through the Doctor who canon and you will find all sorts of contradictions around the number of regenerations. If you include all the works which are not considered canonical there are many more. I could go on boring you with the detail, but I won't as I consider the point to be made. Nobody, not even you, knows how many times the doctor can regenerate.
And I don't see anything posted that would make you say anything was made up here. You must belong to the cult of Forming-an-Opinion-and-Posting-Before-You-Read. Welcome!