Lord President Eternal was a title used for a President of the High Council of the Time Lords who held immortality and thus intended to hold onto the title permanently, rather than planning ever to pass on their power to a successor.

Borusa came to believe that he deserved to be President Eternal, and that his everlasting reign as a benevolent dictator would be beneficial to Time Lord society. This led to him investigating the gift of immortality promised to the winners of the ancient Game of Rassilon in the Death Zone. (PROSE: The Legacy of Gallifrey) However, after he claimed his prize, a projection of Rassilon mockingly revealed to him that the immortality given to the "winners" was that of a living statue forever trapped in the Tomb of Rassilon. Borusa was thus unable to become President Eternal, and, indeed, de facto lost the Presidency, which was passed onto the Fifth Doctor. (TV: The Five Doctors)

After being brought out of the Matrix in a new body, Rassilon himself (AUDIO: Desperate Measures) ultimately claimed the title of "Lord President Eternal" during the Last Great Time War. (AUDIO: Homecoming) Indeed, three subsequent regenerations of Rassilon kept the title of Lord President, (AUDIO: Homecoming, TV: The End of Time, TV: Hell Bent) but the last of these was ultimately deposed by the Twelfth Doctor. (TV: Hell Bent)