Ulysses

An unnamed Time Lord was the father of the Doctor.

Biography
The Doctor's father often discussed wars, great projects, and big things that had to be taken care of with his friends, such as Mr Saldaamir. When the Doctor was "barely a loomling", he let a cobblemouse loose in the great hall of the House of Lungbarrow and it scattered his father's plans and notes in the air, which made him very angry. (PROSE: Unnatural History) Ulysses once watched a meteor storm with his son on Gallifrey, which the latter would remember shortly after his seventh regeneration. (TV: Doctor Who) The Seventh Doctor had a flashback to a time when his father had his arm across the Doctor's shoulders as together they watched a Gallifreyan dawn. (PROSE: Matrix) The Eighth Doctor relived a memory from his first incarnation where Ulysses held him up as a child to see the stars. (PROSE: The Eight Doctors)

A version of history existed where the Time Lords banned all mention of the Doctor's father's name for consorting with aliens, and one where he got every record of himself deleted from the files. (PROSE: Unnatural History)

Other universes
In his time at the Academy, the Doctor's father, like his own father, was tutored by Patience, an old friend of the House of Lungbarrow.

The Doctor's father was also friends with Savar before he went on his mission to find Omega. The Doctor's father chose to stay behind on Gallifrey rather than attempt to rescue Omega from the black hole he was trapped in.

When the Doctor was young, his mother and father went on a trek in the mountains with him. The Doctor's father and mother owned a summer house on the other side of Kasterborous. The Doctor's father was still on Gallifrey when the Doctor had children, and stood at his side while he lost a chess match to Savar.

A mysterious, possibly alternative version of the Doctor was once told that his father's name wasn't Ulysses, and he could be a professor at Berkeley University on Earth in the 21st century. (PROSE: The Infinity Doctors)

Later references
The Eighth Doctor once said that he could remember being Loomed, but also having a father. The Doctor knew one of these was a dream but was unsure as to which. (PROSE: The Shadows of Avalon)

The Eighth Doctor said that he was immortal on his father's side. The sentient quantum field of the planet Albert later told the Doctor that there was a reality where he knew his father's face. (PROSE: Grimm Reality)

Irving Braxiatel said that his father had a cellar full of Draconian brandy. (AUDIO: Everybody Loves Irving)