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.

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 were he described himself as "just a kid".(DW: The Stolen Earth)

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).

He could have been around 236 when he left Gallifrey with Susan (DW: The Pirate Planet)

This statement, as well as one made during his second incarnation (see below) if we take it at face value, make this very possibly the longest-lived of the Doctor's incarnations.

Second Doctor
At some indeterminate time after his first regeneration, 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 unknown when this 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. As established in Time Crash, the Doctor sometimes shows signs of aging as a result of being taken out of time. This would imply that the Doctor and Jamie had their memories wiped of the events of The Two Doctors once returned to their normal time streams. It is also possible, however, that this event may have taken place after the Doctor's trial on Gallifrey (DW: The War Games), but before he began his exile on Earth (DW: Spearhead from Space)
 * 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)

Fourth Doctor
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. However on the last occasion, 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)

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; there is a possibly apocryphal account that approximately two years occurred between the two events (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).

Seventh Doctor
Hours after his regeneration, the Doctor's current incarnation said that he was 953, the same age as the Rani (DW: Time and the Rani). This would mean that approximately 500 years passed between the Doctor's second and sixth incarnations. By the time of his regeneration, he had aged visibly. (DW: Doctor Who: The TV Movie)

Eighth Doctor
While newly with Samantha Jones, at least a few years after his regeneration (having just regenerated, the Doctor took a detour between meeting her and returning for her, which took less than a day from her POV, but longer from his), the Doctor gave his age as 1012. (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 (EDA: Escape Velocity)

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.
 * His comment "900 years of phone-box travel" could be interpreted to mean he doesn't count his life on Gallifrey in his age anymore.

Tenth Doctor
The Doctor described himself as 903 years old (DW: Voyage of the Damned, etc.) River Song compared how young the eyes of the tenth Doctor looked when she met for (from the Doctor's perspective) the first time when contrasted with the version of him she knew earlier in her life but later in his. (DW: Silence in the Library)
 * There is some uncertainty as to whether River Song actually knew the Tenth Doctor and not a later incarnation.

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). Assuming the previous incarnation's claim of 900 is accurate, it's unlikely all the chronicled events involving Rose Tyler (both with the ninth and tenth incarnations) took place in less than a year.

Assuming The Doctor is either lying about his age or ignoring his early life (the first 236 years of his life) on Gallifrey, The Doctor could in fact be at least 1136 when he was with Rose Tyler and 1139, during the Titanic incident.

Behind the scenes
The decision by the writers of the 2005 revival of the series to 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 rationalizations (and fans have suggested others ranging from Time War-related trauma to speculation of an unchronicled adventure in which the Doctor was "de-aged"), to date no episode, novel or comic strip has attempted to address the discrepancy. One simple explaination would be that the Doctor has, taken to lying about his age.