The Doctor's age

Contradictory information has pointed at different estimates of the Doctor's age, both in conversations and in terms of the length of various incarnations.

References and known information
It is unknown what the Doctor's life expectancy is, given his ability to regenerate. In his second incarnation he once stated that as a Time Lord he can "live forever, barring accidents" though it's unclear whether this is accurate or an exaggeration.

Also it seems that Time Lords age in a different way to Humans. The Doctor once said to his companion that he was a Teenager for 50 years. (DWM: The Time Witch)

First Doctor
The Doctor was 8 when he entered the Academy, along with The Master (DW: The Sound of Drums) and 90 when he first visited the Medusa Cascade where he later described himself as having been "just a kid".(DW: The Stolen Earth)

He was a Teenager for 50 years. (DWM: The Time Witch)

He must have been over 90 during the above period.

The Doctor spent "centuries" studying at the Time Lord Academy. (DWM: Mortal Beloved) Magnus chided the Doctor for not regenerating and holding on to this incarnation as long as he did (DWM: Flashback).

This incarnation regenerated at or about age 450. (DW: The Tomb of the Cybermen)

Second Doctor
At some indeterminate time after his first regeneration (but not too long after given the continued presence of Jamie McCrimmon), this incarnation of the Doctor made the first known direct reference to his age. To Victoria Waterfield, he described himself as "something like" 450 years old. (DW: The Tomb of the Cybermen). The Doctor seemed to have aged visibly by the time he went on a mission for the Time Lords to Space Station Chimera (DW: The Two Doctors).
 * It is unclear when these events actually took place within the era of the Second Doctor. He may have been taken out of time during a point prior to The War Games. See Season 6B for a longstanding theory involving the Second Doctor's activities after the trial.

Third Doctor
This incarnation of the Doctor started to say that he had been a scientist for "several thousand" years, but stopped himself before completing the sentence. (DW: The Mind of Evil) He had made a similar age reference shortly before that incident. (DW: Doctor Who and the Silurians). However, he could be referring to the time span he has travelled in, rather than his actual age.

Before his regeneration, he was stranded in his TARDIS for 10 years before landing on Earth and regenerated at 748, making him 738 before he was stranded. (DW: Planet of the Spiders)

Fourth Doctor
It would seem that the Fourth Doctor would have been 748 at the time of his regeneration. (DW: Planet of the Spiders) This incarnation of the Doctor consistently described himself as around 750. He described himself as 749 when he travelled with Sarah Jane Smith (DW: The Brain of Morbius, The Seeds of Doom), 750 with Leela (DW: The Robots of Death) and 756 with Romana during her first incarnation just prior to finding the first segment of the Key to Time. On that occasion, however, Romana corrected him and described him as 759. (DW: The Ribos Operation) Before the conclusion of the quest for the Key to Time, he turned 760 (DW: The Power of Kroll)

On another occasion, the Doctor described himself as possibly 730 and also confessed he couldn't remember his actual age. (DWM: The Time Witch)

Fifth Doctor
The Doctor described himself as 813 when he regenerated (MA: Cold Fusion)

Sixth Doctor
The Doctor described himself as 900 years old (DW: Revelation of the Daleks) and later modified this to be "900 years, more or less". (DW: The Trial of a Time Lord: The Mysterious Planet)

An unspecified gap occurred between these two references, during which his human companion, Peri Brown, visibly matured; one possibly apocryphal account placed the events two years apart. (DWM: The World Shapers)
 * During this story Peri appears as she does at the start of the adventure on Ravolox and mentions to Frobisher that "a couple of years" had passed since she and the Doctor last encountered Jamie McCrimmon and the Second Doctor in The Two Doctors).

At some unspecified point in his past, the Doctor had blown out the candles on his 900th birthday cake. (BFA: The One Doctor)

He was 953 years old when he regenerated. (DW: Time and the Rani)

Seventh Doctor
Hours after his regeneration, the Doctor unlocks a door in the Rani's laboratory with the numerical code "953," which he states is both his age and the Rani's. (DW: Time and the Rani)

The Doctor spent unspecified millions of years in a coma, frozen in a block of ice. (BFA: Frozen Time)

By the time of his next regeneration, he had aged visibly. (DW: Doctor Who)

Eighth Doctor
Three years after his regeneration, the Doctor calculated his total age to be 1012, making him 1009 when he regenerated. He admitted that he was unsure, however, as there was a question of whether he lost count. For this reason he preferred to simply start counting at his most recent regeneration, giving his age as "three." (EDA: Vampire Science)

The Doctor aged while trapped as an amnesiac on Earth in "real time" between the late 19th century (EDA: The Ancestor Cell) and the year 2001, making him at least 1125 by the time this crisis was over. (EDA: Escape Velocity)

At one point while traveling 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. (BFA: Orbis)

Ninth Doctor
This incarnation claimed "900 years of time and space", i.e. travel in his TARDIS, and when asked, said that this was his age. (DW: 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" (DW: The Empty Child). These statements appear to contradict his earlier stated ages.
 * See 'The "900" Controversy' below for an out-of-universe discussion of this change.

Tenth Doctor
The Doctor described himself as 903 years old (DW: Voyage of the Damned, etc.)

While in a parallel universe, he claimed to have given away 10 years of his life to provide the TARDIS with energy. (DW: Rise of the Cybermen).
 * Whether this was a physical aging of the Doctor's body or a sacrifice of expected lifespan off the end of his life remains unknown.

The Doctor's claim of 903 is disputed. As at one point prior to the Titanic incident he is known to have spent at least 2 1/2 years searching for Martha Jones in deep space (DW: The Infinite Quest), as well as several months (at least) living in various time periods on Earth (DW: Human Nature and DW: Blink). He again claimed 903 during another adventure with Donna Noble (BBCR: The Nemonite Invasion).

During a later visit to America, the Doctor claimed an age of 900 once again, suggesting he was rounding down. (DW: Dreamland)

Just prior to regenerating, he stated his age explicitly as 906, suggesting that several years passed for the Doctor after his travels with Donna Noble, a partial accounting of which he gave Ood Sigma. (DW: The End of Time)

Eleventh Doctor
Just prior to his regeneration, the Doctor claimed to be 906. (DW: The End of Time)

Later on, while Amy Pond attempted to seduce the Doctor, he told her that he was 907. (DW: Flesh and Stone)

He again claimed to be 907 when meeting the Dream Lord. (DW: Amy's Choice)

The "900" Controversy
The decision by the writers of the 2005 revival of the series to explicitly describe the Doctor as 900 years old, despite evidence to the contrary on TV and expanded media, is one of the few notable contradictions to established canon in the revival. Although as noted above it's possible to suggest rationalisations (and fans have suggested others ranging from Time War-related trauma to speculation of an unchronicled adventure in which the Doctor was "de-aged" much as he was "aged" in DW: The Sound of Drums and Last of the Time Lords), to date no episode, novel or comic strip has attempted to address the discrepancy. One simple explaination would be that the Doctor has, simply, taken to lying about his age (precedent for this exists in DW: The Ribos Operation in which Romana catches the Doctor shaving a few years off his age - something she, herself, apparently does after her first regeneration). Another is that his measurements of years might vary based on which planet he's using as a baseline; a year on Gallifrey is unlikely to be the same as a year on Earth. Another may be that he has come to base his age off the time he has spent travelling in the TARDIS.

"He Has No Clue"
A simpler and even more self-evident theory is that the Doctor doesn't really know how old he actually is, as constantly traveling backward and forward through time could make it very difficult to measure one's lifespan at all. This theory has been fanon, and the Doctor occasionally admitted to being unsure of his age in spinoff media. (DWM: The Time Witch, EDA: Vampire Science)

This is the theory subscribed to by Doctor Who's current executive producer and head writer Steven Moffat, as stated in an interview with SFX Magazine: "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."

- Steven Moffat

Moffat reaffirmed this on his Twitter blog.