According to dialog in various episodes, the Doctor apparently aged about 450 to 500 years during the course of old Doctor Who in the period 1963 to 1989.
And this is a problem, because the doctor would have overlapping sets of human companions for several seasons at a time, and none of those companions seemed to age while aboard the Tardis.
Furthermore, during the first few seasons dialog indicated that each story or serial happened soon after the previous one. There were many examples of a story beginning right after the previous one ended. So during the early seasons of Doctor Who the total elapsed time aboard the Tardis and at destinations was probably approximately the total duration of fictional time elapsed during the various stories.
Of course the fictional time of an episode or serial might be many times the total onscreen time, but there were few serials where the plot required weeks or months to pass.
Season 12 (1974-1975 reverted to that practice.
All serials in this season continue directly one after the other, tracing one single problematic voyage of the TARDIS crew. Despite the continuity, each serial is considered its own standalone story.
Season 16 (1978-1979) had an overall story arc, and thus there was probably not much time between the individual stories in that season.
Season 16 consists of one long story arc encompassing six separate, linked stories. This season is referred to by the umbrella title The Key to Time and has been released on DVD under this title.
Seasons, 18, 19,and 20 (1980-1983) had overall story arcs which indicated that the episodes happened in relatively quick succession for the passengers in the Tardis. Often one story began immediately after the previous one ended. So it would be hard to imagine the Doctor living for centuries during those three seasons.
In a return to the format of early seasons, virtually all serials from Seasons 18 through 20 are linked together, often running directly into each other. Season 18 forms a loose story arc dealing with the theme of entropy. Full Circle, State of Decay, and Warriors' Gate trace the Doctor's adventures in E-Space; they were released in both VHS and DVD boxsets with the umbrella title The E-Space Trilogy.
So there are considerable problems finding ways for the Doctor to age centuries during the old Doctor Who from 1963 to 1989. But statements about the Doctor's age in various episodes indicates that he did experience centuries of living.
I have a copy of *Who's Next: An Unofficial and Unauthorised Guide to Doctor Who, Mark Clapham, Eddie Robson and Jim Smith, 2005, and I have tried to look up all the statements about the Doctor's age I remember the authors mentioning in the Doctor Who? section of each story discussion, with the details which each story gives about the Doctor.
In "The Tomb of the Cybermen", serial 37, the Second doctor says he is about 400, make it 450, years old (p.96).
Guybrush Mckenzie's answer provides a quote.
Well, if we count in Earth terms, I suppose I must be about four hundred, yes, about four hundred and fifty years old.
In serial 52, "Doctor Who and The Silurians", the Doctor mentions having been in the age of the dinosaurs - which of course ended 66 million years ago - and says that his life has covered thousands of years (p. 129). In Guybrush Mckenzie's answer the quote is:
You know, I'm beginning to lose confidence for the first time in my life. And that covers several thousand years.
In serial 84, "The Brain of Morbius", the Doctor is 749 (p. 200).
In serial 98, "The Ribos Operation", Romana says the Doctor is 759, but he says he's 756 (p. 229).
In Guybrush Mckenzie's answer the quote is:
ROMANA: That's ridiculous for somebody as old as you are.
DOCTOR: I'm not old. What?
ROMANA: Seven hundred and fifty nine?
DOCTOR: Seven hundred and fifty six. That's not old, that's just mature.
ROMANA: You've lost count somewhere.
In serial 142, "Revelations of the Daleks", The Doctor says he is 900 years old (p. 327).
In Guybrush Mckenzie's answer he says that in The Mysterious Planet (1986), part of the season-Long Trial of a Time Lord, the Doctor says:
I may look old to you, whiskerless youth, but I'll have you know I'm in the prime of my life. I'm only nine hundred years old.
And in the next serial, number 144/145, "Time and the Rani", the Doctor says that he and the Rani are the same aged 953 years old.
Peri is the companion of the Doctor in The Mysterious Planet (1986), part of the season-Long Trial of a Time Lord, when the Doctor says he is nine hundred years old.
In "Mindwarp", the part next of the season-Long Trial of a Time Lord (1986) Peri is the Doctor's companion until the Doctor is yanked away from the situation they are in to begin his trial. And she doesn't look older than in "The Mysterious Planet".
In "Terror of the Vervoids", the next part of Trial of a Time Lord (1986), The Doctor shows future events as testimony in his trial. I don't know how the Doctor can remember events in his personal future to think of using them in his defense. Anyway, in those future events the Doctor has a never before seen companion, Mel.
In "The Ultimate foe", the last part of Trial of a Time Lord, The Doctor's future companion Mel arrives in the courtroom, and at the end they leave in the Tardis together.
And in "Time and the Rani", the first episode of the next season, the Sixth Doctor is injured and regenerates into the Seventh Doctor. In a later conversation with another Time Lord, the Rani, it is said they are both 953 years old.
But Mel doesn't look like she aged 53 years between "The Ultimate foe" and "Time and the Rani" like the Doctor says that he did.
And of course there are a bunch of possible theories to explain that.
One) Maybe the Doctor left Mel at some place and time and had about 53 years of adventures in the Tardis and returned to pick up Mel a few minutes after he left her, and thus aged about 53 years more than Mel did during the interval between "The Ultimate foe" and "Time and the Rani".
Two) Maybe Mel, and possibly some or all of the other companions of the Doctor don't age while travelling with him in the Tardis. Thus Mel might have travelled in the Tardis with the Doctor for 53 years without appearing to age. If that is the explanation, and if it applies to every companion, every time a companion of the Doctor decides to quit they are giving up decades or centuries of lifespan. I note that the Doctor never tells quitting companions that they have shortening their life expectancies.
Three) Maybe in The Mysterious Planet (1986), part of the season-Long Trial of a Time Lord, when the Doctor says he is nine hundred years old, he is using years which are somewhat longer than the (posssibly Galifreyan) years used in "Time and the Rani" when he says he and the Rani are 953 years old.
Thus the years in The Mysterious Planet would be about 1.059 times as long as the years in "Time and the Rani", and the years in "Time and the Rani" would be about 0.944 times as long as the years in The Mysterious Planet.
Four) Maybe "Time and the Rani" happens in an alternate universe to Trial of a Time Lord (1986). Maybe in "Time and the Rani" the Doctor saved Peri in the events of "Mindwarp" and she was his companion for some time afterwards. Then the Doctor traveled alone and with other companions in unseen adventures, probably for at least 50 years, before Mel became his companion the way she would have in the timeline of "Terror of the Vervoids". And she is the companion when the sixth Doctor regenerates into the Seventh Doctor in "Time and the Rani" aged 953. And that might be the alternate universe in which the events of the Seventh Doctor's adventures in the last seasons of the old Doctor Who happen.
But in an alternate universe the Doctor is about 900 years old when he and Mel leave the courtroom at the end of Trial of a Time Lord (1986).
In new Doctor Who the Doctor starts out being 900 years old according to Rand al'Thor's question. That fits much better with being in the hypothetical alternate universe of Trial of a Time Lord (1986) than with being in the hypothetical alternate universe of "Time and the Rani".
But a single year doesn't seem long enough to contain the end of the 6th lifetime, the entire 7th and 8th lifetimes, and the beginning of the 9th lifetime, of the Doctor.
Thus it seems likely that new Doctor Who may be in an alternate universe which split off from those of "Time and the Rani" and Trial of a Time Lord much earlier.
Possibly the Fourth, Fifth, Sixth, Seventh, and Eighth Doctors were all severely injured and regenerated into the succeeding incarnations earlier in the new Doctor Who alternate universe than in the alternate universes of "Time and the Rani" and Trial of a Time Lord. And since I read that all of the previous doctors have been seen in various new Doctor Who special episodes, it seems that the Doctor regenerates into the same body forms in the same order in those instances. I can't say whether that is typical of Time Lords or not.
So I wonder which specific events in old Doctor Who previous to Trial of a Time Lord have been depicted, mentioned, or otherwise acknowledged in new Doctor Who, since they would have happened before the divergence in the timelines.
Anyway, I favor the alternate universe explanation for those discrepancies in the Doctor's stated age.