Contradictory information has pointed at different estimates of the Doctor's age, both in conversations and in terms of the length of various incarnations.
Statements and known information
It was difficult to determine the Doctor's life expectancy, given his ability to regenerate. The Second Doctor once stated that as a Time Lord he could "live forever, barring accidents". (TV: The War Games) A Time Lord named Handrel claimed that a single incarnation of a Time Lord could live for around ten thousand years. (PROSE: The Time Lord's Story)
It was also evident that Time Lords aged in a way different from humans. The Fourth Doctor once told his companion that he was a teenager for fifty years. (COMIC: Doctor Who and the Time Witch) He also went through spans of linear time longer than average human lifespans without growing visibly older. (PROSE: Escape Velocity, AUDIO: Orbis, TV: The Impossible Astronaut)
The First Doctor was eight when he entered the Academy, along with the Master. (TV: The Sound of Drums) He apparently left primary school at the age of 45. (PROSE: Shroud of Sorrow) He later described himself as having been "just a kid" when he first visited the Medusa Cascade, at the age of 90. (TV: The Stolen Earth) He said he was a "teenager" for fifty years. (COMIC: The Time Witch) When the Fourth Doctor was 759, (TV: The Ribos Operation) Romana noted the Doctor had been travelling in the TARDIS for 523 years. This would have made the Doctor about 236 when he first 'borrowed' the TARDIS and left Gallifrey. (TV: The Pirate Planet)
This figure was broadly supported by the TARDIS itself. When transferred into a humanoid body, the TARDIS said the Doctor had travelled with it for 700 years. At this point, the Eleventh Doctor was around 909 years old, putting his age when he stole the TARDIS about 200. (TV: The Impossible Astronaut, The Doctor's Wife) During the time Susan Foreman travelled with the Doctor, Ian Chesterton, and Barbara Wright, she knew that despite his apparent maturity, in terms of their own species the Doctor was considered an adolescent. (AUDIO: Here There Be Monsters) He stated he had been travelling in the TARDIS for 60 years when he visited 64, putting his age at 296. (PROSE: Byzantium!) The Doctor spent "centuries" studying at the Time Lord Academy. (COMIC: Mortal Beloved) Magnus chided the Doctor for not regenerating and for holding on to this incarnation as long as he did (COMIC: Flashback). This incarnation regenerated when about 450. (TV: The Tomb of the Cybermen)
The Second Doctor told Victoria Waterfield he was "something like" 450 years old. (TV: The Tomb of the Cybermen) However, he also made claims saying he was several thousand years old, but gave no specific age. He gave his age as 500 shortly before his regeneration. (COMIC: The Brotherhood)
The third incarnation of the Doctor started to say to Professor Kettering that he had been a scientist for "several thousand" years, but stopped himself before completing the sentence. (TV: The Mind of Evil) He had said something similar shortly before that incident. (TV: Doctor Who and the Silurians). He apparently regenerated at the age of 748. (TV: Planet of the Spiders)
The Fourth Doctor seemed to be 748 at his start. (TV: Planet of the Spiders) He gave this number as his age not long after, (PROSE: A Device of Death) and later on he consistently said he was "something about 750", about 757 or 749 when he travelled with Sarah Jane Smith (TV: Pyramids of Mars, The Brain of Morbius, The Seeds of Doom), a further year older; 750 when travelling with Leela (TV: The Robots of Death) and 756 with Romana during her first incarnation just before finding the first segment of the Key to Time. Romana corrected him, saying he was 759. (TV: The Ribos Operation) Before they assembled the Key to Time, he turned 760 (TV: The Power of Kroll) and he was still attributed that age while fighting Skagra. (PROSE: Shada) Another time, he said he was possibly 730, but couldn't remember his actual age. (COMIC: Doctor Who and the Time Witch) He claimed at various times to be anything from 400 to a shade over 1000 years old. (PROSE: Evolution). Romana told the Doctor that after the first couple hundred years, it would be hard to remember your exact age.
The Sixth Doctor said he was 900 years old while travelling with Peri. (TV: Revelation of the Daleks, The Mysterious Planet) At some point in his past, he had blown out the candles on his 900th birthday cake. (AUDIO: The One Doctor) The Malamud noted that, because of the many volumes about the Doctor's life, he must be 20,000 years old. The Doctor didn't contradict them. (PROSE: The Book of My Life) At another time, he was 991 during his travels with Melanie Bush. (PROSE: A Wee Deoch an ...?) He was 953 years old when he regenerated. (TV: Time and the Rani)
Hours after his regeneration, the Seventh Doctor unlocked a door in the Rani's laboratory with the numerical code "953", saying it was both his age and the Rani's. (TV: Time and the Rani) The Doctor had his 1,000th birthday while travelling with Ace and Bernice Summerfield. (PROSE: Set Piece)
Three years after his regeneration, the Eighth Doctor calculated his total age to be 1012. He was unsure and might have lost count, and so preferred to simply begin counting at his most recent regeneration, giving his age as "three" when asked. (PROSE: Vampire Science) The Doctor later stated his age as 1018. (PROSE: Autumn Mist) The Doctor aged while trapped as an amnesiac on Earth in "real time" between the late 19th century (PROSE: The Ancestor Cell) and the year 2001, making him at least 1125 by the end of this crisis. (PROSE: Escape Velocity) While travelling with Lucie Miller, the Doctor spent 600 years on the planet Orbis. He later mentioned that he rounded and adjusted his age based on different year lengths in different parts of the universe. (AUDIO: Orbis)
Shortly before the end of his life, the War Doctor said he was 400 years younger than the Eleventh Doctor when he gave his age as "twelve hundred or something", making him somewhere between 800 and 900 years old. (TV: The Day of the Doctor)
The Ninth Doctor claimed "900 years of time and space", presumably referring to travel in his TARDIS. He also gave this as his age shortly after bringing Rose Tyler back home. (TV: Aliens of London) He later claimed to have used the name "the Doctor" for nine centuries and to have had "900 years of phone-box travel" while with Rose and Captain Jack. (TV: The Empty Child)
The Tenth Doctor said he was 903 years old while attempting to rescue the Titanic. (TV: Voyage of the Damned) Prior to the Titanic incident, he claimed to have spent at least 33 months searching for Martha Jones in deep space (TV: The Infinite Quest), and several months living in various time periods on Earth (PROSE: The Stone Rose, TV: Human Nature / The Family of Blood, Blink) and inside the Valiant during the so-called Year That Never Was. (TV: Last of the Time Lords) He again claimed 903 during an adventure with Donna Noble. (AUDIO: The Nemonite Invasion) In a later adventure in Nevada, he said 900 again, suggesting he was approximating. (TV: Dreamland) While confronting a rabbit in Elizabethan England he briefly believed to be a Zygon in disguise, he said he was 904 years old. (TV: The Day of the Doctor) Just prior to regenerating, he gave his age as 906, suggesting that several years had passed for the Doctor after his travels with Donna Noble. (TV: The End of Time)
Not long after Amy Pond first encountered the Eleventh Doctor, he told her that he was 907 (TV: Flesh and Stone), which he then restated when meeting the Dream Lord. (TV: Amy's Choice) When his future self invited him to Utah, the younger Doctor gave his age as 909. (TV: The Impossible Astronaut) The Doctor later commented that he was around George's age of eight "about a thousand years ago". (TV: Night Terrors) When asked if he knew any tricks, the Doctor stated that he'd been escaping for "about a thousand years, give or take a century or so." (PROSE: Magic of the Angels) When he met Amy, Rory, River and Canton at Lake Silencio, his older self told Amy that he was 1103. (TV: The Impossible Astronaut) Indeed, the Doctor at this point in his life had just finished his farewell tour, which therefore appeared to have lasted about 200 years. (TV: The Wedding of River Song) By the time he had left Amy and Rory, but still offered them brief trips in the TARDIS, he stated his age as 1200. (TV: A Town Called Mercy) When he met Clara Oswald he rounded his age down to 1000. (TV: The Bells of Saint John) The Doctor spent three years trapped in a vast room within the TARDIS. (COMIC: Sky Jacks) When asked by the War Doctor, he said he didn't know and had lost track, settling on "Twelve hundred and something, I think, unless I'm lying." He went on to say that he was so old he couldn't remember if he was lying about his age. (TV: The Day of the Doctor) The Doctor told Theol that he was "nearly" 1500, but he could be a few hundred years off either way. (PROSE: An Apple a Day) He spent 300 years defending the world of Trenzalore, making him at least 1500 years old by the time Clara saw him next. (TV: The Time of the Doctor) By the time the Mara began attacking the residents of Trenzalore, the Doctor had been there for 750 years, making him 1950. (PROSE: The Dreaming) In total, the Doctor spent 900 years on Trenzalore, making him 2100 years old at the end of his first complete cycle of regenerations. (PROSE: Tales of Trenzalore: The Eleventh Doctor's Last Stand)
Behind the scenes
The "900" discrepancy
Before the Eleventh Doctor's 200-year farewell tour between TV: The God Complex and Closing Time, the writers of the 2005 revival of the series explicitly described the Doctor as around 900 years old, despite contrary statements on TV and expanded media. It is one of the few notable contradictions of previously established material in the revival. To date no episode, novel or comic strip has addressed the discrepancy.
"He has no clue"
A simple and self-evident theory is that the Doctor doesn't know how old he is because he has lost count, constantly travelling backward and forward through time making it difficult to measure one's lifespan at all. This theory began as fanon, with the Doctor occasionally admitting to being unsure of his age in spinoff media (COMIC: Doctor Who and the Time Witch, PROSE: Vampire Science); eventually it became solidified as canon in a televised story. (TV: The Day of the Doctor), although the War Doctor's immediate response suggesting he was 400 years younger than the Eleventh complicates matters.
The thing I keep banging on about is that he doesn't know what age he is. He's lying. How could he know, unless he's marking it on a wall? He could be 8,000 years old, he could be a million. He has no clue. The calendar will give him no clues.
The first non-fictional writing about regeneration, The Phoenix in the TARDIS, stated that the First Doctor was about 900 at the time he was replaced by the Second Doctor.