Immortality

Immortality was the ability to live forever, or at least a vastly long time. Most beings called immortal were not also indestructible, but there were exceptions.

Attempts at gaining immortality
Lord President Borusa determined that Rassilon had discovered the secret of true immortality, and used the first five incarnations of the Doctor to obtain it. This turned out to be the immortality of a living statue. (DW: The Five Doctors)

According to Iris Wildthyme, she defeated Morbius's attempt to obtain Rassilon's gift of immortality. (EDA: The Scarlet Empress)

Queen Xanxia of Zanak attempted to create an immortal body for herself using the energies stolen from other planets. (DW: The Pirate Planet)

Mawdryn and his group of scientists stole a metamorphic symbiosis regenerator from the Time Lords and used it on themselves to gain immortality, only to find that theirs was one of unending mutation and regeneration. (DW: Mawdryn Undead)

Astrolabus, who was in his final incarnation, attempted to gain immortality by stealing Voyager's star charts. (DWM: Once Upon a Time Lord)

Qataka, afraid of death and having heard stories of the Time Lords, experimented with cybernetics in an attempt to gain immortality. (NA: Timewyrm: Genesys)

Professor Richard Lazarus invented a device which would reverse the ageing process, providing a form of immortality, but it also mutated him into a monster. (DW: The Lazarus Experiment)

The Family of Blood pursued the Tenth Doctor in an attempt to gain his body, thus becoming immortal. As punishment for their crimes, the Doctor granted them immortality in various unpleasant ways. (DW: The Family of Blood)

Joshua Naismith attempted to use the Vinvocci Immortality Gate to make his daughter immortal. (DW: The End of Time)

Angelo Colasanto attempted to make himself immortal after seeing Jack Harkness die and come back to life. (TW: End of the Road)

Gained immortality
The Cybermen gained immortality by replacing their body parts with cybernetic ones, but they lost their emotions and humanity in the process. (DW: The Tenth Planet)

The race known as the Olympians had the power of immortality. (DWM: The Life Bringer)

Jack Harkness, after being revived by the Bad Wolf, would always come back to life after being killed, and his ageing was slowed down. (TW: Everything Changes) He lost this immortality on Miracle Day, (TW: The New World) but regained it when the Miracle was ended. (TW: The Blood Line)

Suzie Costello, after being brought back to life, was unable to die again until the Resurrection Gauntlet that was keeping her alive was destroyed. (TW: They Keep Killing Suzie)

Katsura Sato gained immortality after having been rescued by Nanodrones (DWM: The Road to Hell), but lost it when Kroton took control of the Glory. (DWM: The Glorious Dead)

Thomas Vaughan discovered the secret of immortality in the mid 17th century. (TWA: Hidden)

In an alternate timeline, the Nimon granted immortality to Sebastian Grayle. (BFA: Seasons of Fear)

The Warlord Grayvorn gained immortality from the Relic. (BFA: Excelis Rising)

Erasmus Darkening gained immortality. (SJA: The Eternity Trap)

In 2011, during an event dubbed by the world press as "Miracle Day", the human race stopped dying. (TW: The New World) This was eventually reversed. (TW: The Blood Line)

Rex Matheson gained immortality through unknown means. (TW: The Blood Line)

The disembodied heads of the members of the Order of the Headless lived on, even when they were reduced to skulls. (DW: The Wedding of River Song)