The Doctor's age

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

First Doctor
Magnus chided the Doctor for not regenerating and holding on to this incarnation as long as he did (DWM: Flashback). This statement, as well as one made to Victoria Waterfield (see below), make this very possibly the longest-lived of the Doctor's incarnation.

Second Doctor
In his second incarnation, at some indeterminate time after his first regeneration, 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; aired during Patrick Troughton's second season). The Doctor aged visibly in the time between his trial on Gallifrey (DW: The War Games) and his going on a mission for the Time Lords, in which he visited Space Station Chimera (DW: The Two Doctors)

Third Doctor
This incarnation of the Doctor described himself as many thousands, not hundreds of years old. (DW: Doctor Who and the Silurians, DW: The Mind of Evil)

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)

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 the younger age of only 900. (DW: Aliens of London) But then he also claimed "as he states "900 years of phone-box travel", i.e. travel in his TARDIS (DW: Aliens in London).

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) for 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) The Doctor doesn't appear to be taking into account three years spent searching for Martha Jones (DW: The Infinite Quest).
 * There is some uncertainty as to whether River Song actually knew the Tenth Doctor and not a later incarnation.