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)
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 confirms 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.