|This page needs a major clean up or extra information added because
You can clean up this page by hitting edit above. Once this page has been cleaned, please remove this template from the page.
As of the end of Series 7 (2013), the answer to this is: unknown.
In the 1976 story "The Deadly Assassin", it was established that Time Lords can regenerate 12 times for a total of 13 "lives." This piece of backstory is introduced by way of explaining the change in appearance of the Master, seen here trying to cling to life at the end of his 13th incarnation. The Master eventually gets around this by taking over the body of Tremas of Traken in "The Keeper of Traken", and later Bruce the paramedic in "Doctor Who", the 1996 TV movie.
Several storylines reference the 13-life limit. "Mawdryn Undead" and the 1996 TV movie both base their plots around characters trying to steal the Doctor's remaining regenerations. "Shada" (unbroadcast) and "The Twin Dilemma" both discuss Time Lords reaching the end of their lives.
However, on at least three occasions it has been clearly shown that regeneration limits were imposed as opposed to being natural. In "The Five Doctors", the Master is offered a new regeneration cycle by Lord President Borusa; in the 1996 TV movie the Doctor's voiceover at the start refers to the limit as a "rule" that didn't mean much to the Master; in "The Sound of Drums" it is stated that the Time Lords granted the Master new regenerations in return for him helping out in the Time War.
"The End of the World", et al, established that the Time Lords no longer exist (their appearance in "The End of Time" being a universe-warping anomaly caused by outside interference). It is not known if this means there is no longer any control over the number of regenerations. In the "Death of the Doctor" (Sarah Jane Adventures story), when asked, the Doctor states he can regenerate 507 times. The writer of the episode, Russell T Davies, has stated that this was meant to be a joke. On screen, this is less clear. (Note: some online sources erroneously state that the Doctor claims to have no limit; this is likely based upon some erroneous media reviews of the episode, plus the fact the Doctor states that there is no limit as to what he could look like after a regeneration when asked by Clyde Langer if he could be black).
There have been other factors that call into question whether the Doctor is subject to any regeneration limits. In "Let's Kill Hitler", for example, River Song transfers her remaining regeneration energy into the Doctor to save his life. Only two regenerations are known for River, so she could have potentially given the Doctor 10 or 11 more. However, in "The Angels Take Manhattan", the Doctor gives some of the energy back to River; the circumstances of her death in "Forest of the Dead" makes it impossible to speculate whether she regained the ability to regenerate as a result.
Complicating matters further are events of four episodes that call into question the actual number of lives the Doctor has had, or will have. In "The Brain of Morbius" we see a number of unknown faces that are implied to be pre-Hartnell incarnations of the Doctor; the season-long story arc "The Trial of a Time Lord" introduces the Valeyard as a future incarnation of the Doctor from sometime between his 12th and final regeneration - note no firm number is given); in "The Stolen Earth"/"Journey's End", the Doctor undergoes an aborted regeneration which left fans wondering if it was part of the count; in "The Name of the Doctor", we are introduced to a previously unknown incarnation of the Doctor, whose placement within the chronology of the Doctor's life remains, for now, unknown. For her part, in "Name of the Doctor" Clara clearly states that there have been 11 Doctors to this point, which at least would seem to debunk the "Morbius Doctors" claim. All this refers to TV only; the novels have suggested other scenarios for the Doctor.
In real life, people involved with the production of the series have been openly dismissive of the limit. This includes Russell T Davies, Steven Moffat and David Tennant. Simply put, it is not possible to even speculate at this point whether any regeneration limit still exists for the Doctor.