As of the end of Series 10 (2017), this is unknown.
In "The Deadly Assassin", it was established that Time Lords can regenerate 12 times for a total of 13 "lives." In "The Five Doctors", however, it was established that the Time Lords could grant a new regeneration cycle but we learned nothing about how this could be done, nor did we learn how many additional regenerations it would allow. (The Master was offered a new regeneration cycle but never actually given one.)
In "The Five Doctors", too, we learned from Borusa that Rassilon had achieved "Timeless perpetual bodily regeneration. True immortality!"
In "Utopia", we found out that the Master had been resurrected by the Time Lords during the Last Great Time War, as they thought (wrongly, as it turned out) that he'd make a good warrior. He had regained his ability to regenerate but, again, we gained no information about how many times he could do so.
In "The Name of the Doctor", we discovered that there had been an incarnation of the Doctor (the "War Doctor"), & therefore a regeneration, we hadn't previously known about. That regeneration was shown in "The Night of the Doctor", when the Eighth Doctor regenerated into the "War Doctor".
In "The Time of the Doctor", it was confirmed that the Eleventh Doctor was the final incarnation of the Doctor's original regeneration cycle. This confirmed several previously uncertain points:
- The aborted regeneration seen in "The Stolen Earth"/"Journey's End", in which the Tenth Doctor healed himself but didn't change form, did use up one of the Doctor's 12 regenerations.
- That aborted regeneration, plus the previously unknown regeneration of the Eighth Doctor into the "War Doctor", plus the regeneration of the "War Doctor" into the Ninth Doctor (at the end of "The Day of the Doctor"), plus the other regenerations we already knew about accounted for all the Doctor's incarnations.
- The unknown faces seen in "The Brain of Morbius" were not, therefore, earlier incarnations of the Doctor (as some had suspected they might be) & the incarnation known as the First Doctor really was the first incarnation ("The original, you might say," as he himself did say in "The Five Doctors") & there cannot be any other unknown earlier incarnations of the Doctor.
- The Doctor didn't receive any regenerations from River in "Let's Kill Hitler". Her regenerations were expended in healing him, not transferred to him.
In "The Time of the Doctor", as the Eleventh Doctor was finally dying of extreme old age, he was granted "A whole new regeneration cycle," by the Time Lords -- to the extreme consternation of the Daleks, who witnessed the event. In that episode, the Doctor underwent "Regeneration number thirteen" & used the regeneration energy to inflict severe damage on the Daleks, as well as to transform himself into the Twelfth Doctor.
In "Kill the Moon", the Twelfth Doctor (the first incarnation of the Doctor's new regeneration cycle) said that, for all he knew, he might now be able to regenerate without limit. In "Hell Bent", Rassilon, who does himself have unlimited regenerations, was likewise unsure how often the Doctor could now regenerate.
At the end of "Twice Upon a Time", the Doctor regenerated for the fourteenth time, into a female incarnation, the Thirteenth Doctor.
Thus, we know that the Doctor has regenerated fourteen times but the numbering of the incarnations is out of step with the number of regenerations, because the "War Doctor" didn't call himself the Doctor & the Tenth Doctor once regenerated without changing form. We do not, however, know whether or not there is now any limit to the number of times the Doctor can regenerate.