Why does the Doctor never meet someone who's already met him before?

Question

Why does the Doctor (almost) never meet someone for the first time (in his timeline) who has already met him in his future? Especially since he hangs about late 20th/early 21st century London so much ...

Answer 1

He does (sometimes).

The most notable example of this is River Song. They keep on meeting each other time after time, almost never in the right order. Their timelines are closer to being opposite than parallel. For instance, take their very first (from his point of view) meeting:

RIVER: You're doing a very good job, acting like you don't know me. I'm assuming there's a reason.
DOCTOR: A fairly good one, actually.
RIVER: Okay, shall we do diaries, then? Where are we this time? Er, going by your face, I'd say it's early days for you, yeah? So, er, crash of the Byzantium. Have we done that yet? Obviously ringing no bells. Right. Oh, picnic at Asgard. Have we done Asgard yet? Obviously not. Blimey, very early days, then. Whoo, life with a time traveller. Never knew it could be such hard work. Look at you. Oh, you're young.
DOCTOR: I'm really not, you know.
RIVER: No, but you are. Your eyes. You're younger than I've ever seen you.
DOCTOR: You've seen me before, then?
RIVER: Doctor, please tell me you know who I am.
DOCTOR: Who are you?

_{-- Silence in the Library (Series 4 Episode 8)}

Another example is Queen Elizabeth the First:

DOCTOR: Queen Elizabeth the First!
ELIZABETH: Doctor?
DOCTOR: What?
ELIZABETH: My sworn enemy.
DOCTOR: What?
ELIZABETH: Off with his head!
[...]
MARTHA: What have you done to upset her?
DOCTOR: How should I know? Haven't even met her yet. That's time travel for you. Still, can't wait to find out. That's something to look forward to.

_{-- The Shakespeare Code (Series 3 Episode 2)}

In fact, this may be a pattern with women he later ends up marrying - the first time he meets them in his timeline is the last in theirs - to judge from what he says to Sally Sparrow (who is a third example [though he hasn't married her, at least as far as we know], since their first meeting in his timeline is right at the end of the episode):

DOCTOR: Look, sorry, I've got a bit of a complex life. Things don't always happen to me in quite the right order. Gets a bit confusing at times, especially at weddings. I'm rubbish at weddings, especially my own.
SALLY: Oh, my God, of course. You're a time traveller. It hasn't happened to you yet. None of it. It's still in your future.
DOCTOR: What hasn't happened?

_{-- Blink (Series 3 Episode 10)}

Another example is Lorna Bucket, who remembers the Doctor from her childhood in the Gamma Forests, although his first encounter with her is when she dies at Demon's Run:

LORNA: I met you once, in the Gamma Forests. You don't remember me.
DOCTOR: Hey, of course I remember. I remember everyone. Hey, we ran, you and me. Didn't we run, Lorna?
(Lorna dies.)
DOCTOR: Who was she?
VASTRA: I don't know, but she was very brave.

_{-- ^{(one of the most tear-jerking moments of)} A Good Man Goes to War (Series 6 Episode 7)}

Going back to Old Who, there's also the Sixth Doctor companion Mel Bush. To quote from her Wikipedia page:

Mel first appears in the serial Terror of the Vervoids, part of the 14-part story The Trial of a Time Lord. At this point, she and the Sixth Doctor have been travelling together for some time. The events of Vervoids are shown as part of a Matrix projection of future events being shown by the Sixth Doctor to the court, so from his point of view, he is seeing an adventure he will have with Mel even before he meets her in his own timeline. At the end of Trial, the Sixth Doctor leaves with this future Mel, presumably to drop her off somewhere, meet her past self for the first time (from her point of view), and then carry on from there.

Answer 2

TL,DR
The place he hangs about, London, ranges in population from 1 million to nearly 9 million people in the time(s) that he visits. Add the fact that mixing different times into the equation basically compounds the number of people who wouldn't know him, and his relatively young age, it makes sense that he isn't running into people he knows personally constantly.

Long Version

Remember that while the Doctor does time travel, he's not very old. He's rather young compared to the age of the universe, and so he can only be have been in so many places not only physically, but in time, in only a couple thousand years at most. He hasn't been time traveling his entire life either. How many people could you get to know across all human history, and all of Earth in 900 years or so? Not so many that everywhere you go people will know you personally.

However, you specifically mention London:

Especially since he hangs about late 20th/early 21st century London so much ...

The population of London in the time(s) that the Doctor frequents it ranges from 1 million to nearly 9 million people. Remember that he's not just hanging about in one physical place, but the same place spanning a couple hundred years. Given that each new generation is roughly 30 years apart, adding in time travel, it's kind of like visiting 6 different places with a population between 1 and 8.5 million.

Answer 3

Out-of-universe:

To keep things simpler

Since I feel it adds to the explanation, I will consider an out-of-universe perspective.

The show Doctor Who has been around for a very long time. Having such a long running presents a unique set of challenges. It's unrealistic to expect every new fan to spend loads of time and money to go back and catch up on the whole show so they can follow every plot line and reference, so Doctor Who tries to take care to keep many episodes and storylines as mostly self-contained plots.

For example:

• Once the 10th Doctor Regenerated into the 11th, the show rarely if ever makes reference to stories or characters from the 10th Doctor's run. The 11th Doctor brought newer fans to the show, and having too many references to the previous Doctor could make the show less accessible to new fans. When they do choose to return to a pre-existing character or plot line, it's treated as a sort of special occasion and care is taken to briefly refresh our memories in order to help new fans get up to speed.

This also keeps things simpler for writers. Not all episodes are written by the same person and since Doctor Who has no canon, writers are given freedom to disregard pre-existing characters and storylines in favor of newer characters in these often self contained stories. See also Which Dr Who works are canon?

Answer 4

He does occasionally, but to do so long term would require significant planning on the behalf of future storywriters and editors.

Generally it's because it is (was) a low budget childrens' show intended to teach real history in an interesting way - hence the "fixed points in time and space" which is code words for "history that we teach in UK schools". the showed gained popularity as it included technology (science fiction) and natural philosophy in it's story values. (as well as excellent acting). It's also why The Doctor is supposed to be a strange but authoritive figure (hence professional name "the doctor"), slightly weird (adults are always weird alien minds to children), and fun (includes easy to copy and include in childrens play). Thus The Doctor always has to have that special feeling of being up past your bedtime watching tv/reading under the covers when you're not supposed to.... ie it's spicy, it's exciting, and if you're not careful you'll learn something!