Immortality

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

Attempts at gaining immortality
The Time Lords had a form of immortality in their ability to regenerate. Lord President Borusa, however, determined that Rassilon had discovered the secret of true immortality, and used the first five incarnations of the Doctor to obtain it. However, 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)

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

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

The Family of Blood pursued the 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)

Gained immortality
The Cybermen gained immortal bodies but lost their emotions and humanity. (DW: The Tenth Planet)

Jack Harkness, after being revived by Bad Wolf, would always come back to life after being killed, and his aging was slowed down. (TW: Everything Changes)

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)

Erasmus Darkening gained immortality.