The Doctor, according to several accounts, had a father.
Biography
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. (PROSE: The Infinity Doctors)
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. (PROSE: Unnatural History) 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. (PROSE: The Infinity Doctors)
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) The Eighth Doctor recalled his original incarnation lying in the grass as they watched a meteor storm together. (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 his father held him up as a child to see the stars. (PROSE: The Eight Doctors)
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. (PROSE: The Infinity 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) The Doctor was once told that his father's name was not Ulysses, but he could be a professor at Berkeley University on Earth in the 21st century. (PROSE: The Infinity Doctors)
Ultimately, the Doctor chose to think of the universe as his foster family, after his parents "decided to opt out of their responsibilities." (PROSE: Beltempest)
Later references
Maris' potential engine gave several different origins for the Doctor. In one of them he had a Time Lord father and a human mother. (PROSE: Celestial Intervention — A Gallifreyan Noir)
The First Doctor had a dream whilst in the TARDIS of his father sending him out into a body of water in a leaky boat. The image of his father was distorted in the dream, resembling that of a stuffed bear. When the Doctor woke he found he was holding HiFi, which appeared to have distorted the dream. (PROSE: The Three Paths)
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)
When Quintus asked the Tenth Doctor not to tell his dad Lobus Caecilius that he showed the Doctor where Lucius Petrus Dextrus lived, the Doctor responded, "Only if you don't tell mine." (TV: The Fires of Pompeii)
The Saxon Master, holding the Tenth Doctor and Wilfred Mott prisoner, commented, "Your dad's still kicking up a fuss," when Wilfred protested in defence of the Doctor, he replied that he would be proud if he was. Later, the Doctor told Wilfred that he would be proud if he was his father. (TV: The End of Time)
Behind the scenes
Ulysses
In Philip Segal's proposed late 1990s revival of Doctor Who, the Doctor's father was Ulysses. Ulysses appeared in PROSE: The Gallifrey Chronicles as the husband of Penelope Gate, and they were said to have a half-human, half-Time Lord son on Gallifrey. The Eighth Doctor said in TV: Doctor Who that he was half-human on his mother's side.
Penelope Gate was also implied to be the Doctor's mother in PROSE: The Infinity Doctors, where the Doctor knows he can find his father as a professor at Berkeley, but that his name isn't Ulysses. In PROSE: Unnatural History, the Eighth Doctor meets a renegade Time Lord calling himself "Daniel Joyce" as a professor at Berkley. The Doctor says the pseudonym fits him; in real life, the author James Joyce wrote the novel Ulysses. Ulysses was last seen in PROSE: The Gallifrey Chronicles in the company of Larna, who was Joyce's assistant in PROSE: Unnatural History.