The Doctor's new regeneration cycle

The Doctor was granted a new regeneration cycle, (TV: The Time of the Doctor) also called a new life cycle, (PROSE: Regeneration) at the end of the Eleventh Doctor's life, allowing them to live beyond the First Doctor's regeneration cycle. (TV: The Time of the Doctor)

Origins
The High Council only gained the ability to grant new regeneration cycles from the research of Ophiuchus. The Doctor would attribute the existence of their new regeneration cycle to the Fifth Doctor's encounter with Ophiuchus. (COMIC: Ophiuchus) According to another account, the means to grant new regeneration cycles came from the brain-print of Artron, one of Rassilon's great engineers. (AUDIO: Day of the Master)

The Eleventh Doctor was originally to be the Doctor's final incarnation. On the verge of dying of old age during the siege of Trenzalore, the Doctor's companion Clara Oswald pleaded with the Time Lords through a crack in the universe to change the future and save the Doctor. In response, the Time Lords gave the Doctor a new life cycle by sending him regeneration energy through the crack. (TV: The Time of the Doctor)

The Cycle's extent
When Captain Lundvik threatened to shoot him, the Twelfth Doctor speculated that he would "keep on regenerating forever" if he was executed. (TV: Kill the Moon) Rassilon would later state his own uncertainty to the number of regenerations the Doctor had been granted, (TV: Hell Bent) with the Saxon Master and Missy also considering their own uncertainty when debating whether to throw the Twelfth Doctor off a roof, believing they "could [be] up and down the stairs all night." (TV: The Doctor Falls)

The Thirteenth Doctor remarked to Cleo Proctor that she would stand up again if she "clobber[ed]" her, remarking that she would regenerate "Possibly several times" and then "who can say?" as to how many times. (AUDIO: Salvation)

The Curator, an incarnation in the future relative to the Eleventh Doctor, (TV: The Day of the Doctor, COMIC: The Then and the Now) stated that his number was "extraordinarily high". (AUDIO: Lost Property)

Used regenerations
Receiving the new regeneration cycle caused an explosive "reset" in the Eleventh Doctor's body which restored the him to his youthful form and destroyed the attacking Dalek forces. From here, the Eleventh Doctor's regeneration occurred shortly afterwards. (TV: The Time of the Doctor) Eventually, this too was followed by the Twelfth Doctor's regeneration into who they understood would become known as the Thirteenth Doctor, though "those in the know" would understand that she was not truly the thirteenth. (TV: Twice Upon a Time, COMIC: Ophiuchus)

The Twelfth Doctor was forced through seven false regenerations by a "regeneration vampire". With the assistance of the Eleventh Doctor, the regeneration energy he expended was returned to him. (AUDIO: Regeneration Impossible)

The Spy Master subjected the Thirteenth Doctor to a forced regeneration which he used to take control of her body, which changed to reflect his form. The Doctor, however, persisted in the Edge and was eventually restored by regeneration energy taken from the CyberMasters. Soon after, the Doctor was mortally wounded by the Master, leading to her regeneration into the Fourteenth Doctor. (TV: The Power of the Doctor)

Other information
The Vortex Butterfly once explained to the Tenth Doctor that due to his overall timeline's shape as a circle in the Ocean of Time, the Doctor was not "limited" to "thirteen lives." (COMIC: Vortex Butterfies)

Behind the scenes
Taking cue from The Time of the Doctor's wording, various Twelfth Doctor material released in 2014 refer to this regeneration cycle as “a new regeneration cycle”, including the instructions to LEGO 21304 Doctor Who, the author’s notes for The Crawling Terror, Doctor Who The Official Annual 2015, and various issues of Doctor Who Magazine.