The Doctor is from far in the Master's future, and says his age is four billion. Ohila had previously mentioned the Doctor had spent four and a half billion years inside the confession dial. (TV: Hell Bent) However, the Doctor would later describe his age as 2,000. (TV: Smile, et. al) The most likely explanation is that the Doctor is exaggerating for dramatic effect.
The idea of the 'Delgado' Master regenerating here would seem to contradict other sources such as Legacy of the Daleks and The Two Masters, which also apparently depict this version of the Master regenerating. However, since the Master is not explicitly shown in his next incarnation, he may have managed to maintain enough control of his new self to maintain a similar physical appearance to allow for the events of Legacy, and Two Masters includes a scene where the younger, pre-burned Master meeting his future self and the older Master comments that he almost forgot what he looked like before the burns, suggesting that the burned incarnation is not just the disfigured Delgado Master but a new incarnation altogether.
For the purposes of this list, a "regeneration story" is one in which a regeneration is actually and initially depicted. For this reason The War Games is not included below, even though it is commonly thought of as a "regeneration story". It doesn't actually include a clear scene of regeneration, and the preponderance of stories in other media confirm that the Second Doctor did not regenerate at the end of it. Additionally, immediate post-regeneration stories, like the 2005 Children in Need Special — and ones like Castrovalva, where the regeneration sequence was replayed — are not included.
Regeneration is usually considered a biologic process exclusive to Time Lords and a few other species. However, some consider Regeneration and The Eclipse of the Korven to also be "regeneration stories", though neither describes anything close to a biological process. Korven is a particularly hard one to include in this list, because K9 is said to "regenerate" but is largely physically unchanged.