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We don't know how the Doctor and the Master went from school chums to frenemies.
What led him to steal the TARDIS (other than general wanderlust)? There were "pressing reasons at the time", apparently, but good luck getting him to admit what they were.
The Doctor's original family. Susan's the only (confirmed) one we've seen; what about the others? Does it have anything to do with why he left Gallifrey in the first place?
The Doctor frequently references various events in Earth's timeline, such as the "Twelfth British Empire", "The Nineteenth Reich", and several future World Wars.
An early example is from "The Edge of Destruction". Susan recognises a photograph on the screen as Quinnis, a planet in the fourth universe where they nearly lost the TARDIS. As revealed in the Big Finish story "Quinnis", it was almost carried away by a flood when the Doctor was posing as a rainmaker.
He says Gilbert and Sullivan told him he looked better in a cape.
In ''The Name of The Doctor'', the Second Doctor is shown having an adventure on a palm tree-laden beach of contemporary Earth, apparently with the Eighth Doctor.
Similar to Zodin, there been several explanations outside the TV series about what the Perigosto Stick the Doctor mentions in "The Green Death". You can't trust a Venusian Shanghorn with one, you see.
In one episode, it's mentioned that the Brigadier once somehow earned the gratitude of a woman named Doris; during the seventh Doctor's run, it's revealed "she finally caught him" and they are now married. (A comic explains this.)
The "daisiest daisy" monologue is an extremely rare glimpse into the Doctor's childhood on Gallifrey (a house halfway up a mountain, near where an old hermit lived under the tree). The events he describes took place on 'the blackest day of my life' - but he never actually mentions why he felt so depressed. As of "Listen", however, we may have been given a clue - fear of being forced to join the army, since no-one thought he'd make a Time Lord.
In "The Sunmakers", the PA announces that the Gatherer has offered 5,000 telmars for the Doctor's capture. A couple of workers marvel at this, until the Doctor (who's been standing behind them) scoffs, "Peanuts! The Droge of the Gabriellides once offered an entire star system for my head!"
The Doctor's already quite familiar with Leonardo da Vinci when he goes back in time to meet him in "City of Death".
For the entirety of his tenure, the Doctor wears a stalk of celery on his left lapel, pinning it on in "Castrovalva". In his final serial, "The Caves of Androzani", we learn this incarnation suffers allergies from Praxis gasses in star systems. If he comes in range of these gases, the celery turns purple and he eats its to (presumably) act as an immunity booster to the allergens. However, we never see this happen on screen, because this was a last minute explanation written in to justify his habit of pinning the celery to his clothing.
The serial "Timelash" references an unbroadcast adventure the Third Doctor and Jo Grant had on the same planet.
There's also a humorous example:
Peri: You even managed to burn dinner last night! Doctor: I never said I was perfect! Peri: If you recall, I was going to have a cold dinner last night.
His worryingly-throwaway line about the Hand of Omega in "Remembrance Of The Daleks" — "...and didn't we have trouble with the prototype..."
If the series had continued on, this would have been part of then-script editor's Andrew Cartmel's "master plan" to introduce more mystery to the Doctor's character, which is explained in depth in his final Virgin New Adventures novel, Lungbarrow. Apparently, the Doctor is in some way a heir or reincarnation of the Other, one of Gallifrey's founders, who worked with Rassilon and a not-yet insane Omega to construct the Hand of Omega. This is foreshadowing of What Could Have Been on the TV series, where Seven was going to skate further into becoming The Chessmaster.
He and the Master indicate a half-human side with this new incarnation. Writing-wise, this bold move is so much of a Continuity Snarl people try to Hand Wave it or ignore it altogether. The Doctor has always been portrayed as pure Time Lord. Word of God says if the movie got a series, it would be explained... somehow.
In ''The Name of The Doctor'', the Eighth Doctor is shown having an adventure on a palm tree-laden beach of contemporary Earth, apparently with the Second Doctor.
The Doctor knew Madame Curie... intimately.
Invoked by the Master, who implies a terrible backstory for the Doctor to convince Chang Lee to work with him.
The Master: Genghis Khan? Chang: What about him? The Master: That was him! Chang: No way! The Master: Yes way.
The Eighth Doctor was also fond of using Retroactive Precognition to create these, alluding to other characters' future deeds.
2005 revival series:
The Steven Moffat era has made Noodle Incidents a Running Gag.
The Time War that destroyed Gallifrey and (supposedly) the Daleks was treated in this fashion for eight years; we were given lots of intriguing hints and abstract reveals as to what happened and who did what (with mentions of the Fall of Arcadia and the burning of the Cruciform — neither of which were seen in the series — and it being stated in "Dalek" that the Doctor had no choice but to do the deed himself), but we didn't see what actually occurred, though "The End of Time" explains some of it. The Fiftieth Anniversary special "The Day of the Doctor" finally revealed more of the backstory, including the Fall of Arcadia, but other parts such as the burning of the Cruciform remain unexplained.
From "Boom Town": "I told you we should have turned left!" (A phrase which takes on a deeper meaning a couple of series later, quite possibly on purpose, given the trend for significant words and phrases throughout the new series.) We see Jack telling part of a story about a bunch of naked Time Agents and something with tusks.
Also, in "Rose" the Doctor mentions (about the TARDIS' doors) that "The assembled hordes of Genghis Khan couldn't get through those doors. Believe me; they've tried."
Played for Drama in the two-parter "The Impossible Planet"/"The Satan Pit", in which something claiming to be the Devil keeps taunting the astronauts with snippets of their pasts. He repeatedly needles the chief of security about an incident which it claims his wife never forgave him for, suggests the head scientist is "running from daddy", etc.
The Doctor's laser spanner was stolen by Emmeline Pankhurst, according to "Smith and Jones". Cheeky woman!
"The Shakespeare Code" ends with one, when Queen Elizabeth shows up, recognizes the Doctor, declares that he is her "sworn enemy", and has arrows fired at him. As the Doctor hadn't met her (yet) he had no idea what was going to happen. "The End of Time Part 1" clarifies this a bit: Apparently he married her slightly before that episode. The event in question is finally shown in "The Day of the Doctor".
The ending includes some intriguing reasoning for why the Doctor can't stay long.
The Doctor: Got to go, got a thing. Well, four things. Well, four things and a lizard.
Relatedly, the Doctor saying he's "rubbish at weddings," especially his own. And this is before the Queen Elizabeth I incident!
It's been outright stated a few times that we haven't seen anywhere near the number of actual places each companion goes with the Doctor in the episodes themselves. Mostly this is because nothing exciting happened those times (like one flashback from the beginning of "Army of Ghosts") but the Children in Need special "Doctor Who: Children in Need", with the Tenth Doctor convincing Rose he's actually the Doctor, did include him referring to a time they ended up "hopping for our lives."
Events of the show are regularly presented as noodle incidents to other people In-Universe. For instance, when Agatha Christie says in "The Unicorn and the Wasp" that her husband left her for a younger woman, Donna replies "Well, mine left me for a giant spider." It makes sense if you saw the episode, but to Agatha Christie it sounds ludicrous.
"Silence in the Library" plays with this trope by invoking a little "future nostalgia": River Song mentions several escapades ("Right, picnic at Asgard. Have we done Asgard yet?") which are supposed to have happened in the Doctor's future. One of these was shown in the following series — their adventure on the Byzantium in "The Time of Angels"/"Flesh and Stone".
In "The Waters of Mars", Ed did something for which Adelaide "never could forgive" him. Russell T. Davies says in the background interviews that he likes hinting at this mystery that will never be solved.
During "The End of Time", The Doctor references a number of monsters/weapons that fought in the Time War. The names he drops are The Nightmare Child, the Skaro Degradations, the Horde of Travesties, the Could've Been King, and his army of Meanwhiles and Neverwheres. The implication is that each is some kind of Eldritch Abomination but we've never given any indication what they were or what side (if any) they each fought on. Even when the Time War is shown in "Day of the Doctor", none of them seem to appear. The only other reference is in "The Stolen Earth" when the Doctor notes that Davros' command ship flew into the jaws of the Nightmare Child, which points to it being MASSIVE.
Likewise, The Master says in "The Sound of Drums" that we went AWOL during the Time War when the Daleks took control of the Cruciform because he was so afraid. We're never shown what the Cruciform is. Word of God states that it's NOT the same thing as the Crucible, a massive space station the Daleks use in the Series 4 finale. Given the Master's lack of fear against the Doctor, this act must have meant something dire if the Master decided to escape to the end of the universe and hide out as a pathetic human.
The Doctor: That's okay, 1580. Casanova doesn't get born for 145 years. Don't want to run into him. I owe him a chicken. Rory: You owe Casanova a chicken? The Doctor: (briskly) Long story, we had a bet.
From the same episode, the Doctor jumps out of Rory's stag party cake, and when he spots Rory:
The Doctor: That's a relief, I thought I burst out of the wrong cake. (beat) Again.
Eleven has to unlive his whole life as the universe prepares to restart itself, thereby erasing him from existence. He sees Amy and his earlier self entering the TARDIS, Amy agape about "Space Florida" and its "automatic sand".
According to this episode, he accidentally got engaged to Marilyn Monroe in 1952.
Also, there is apparently a moon made of honey. Except it's not actual honey, nor is it a moon. It's also technically alive and a bit carnivorous.
River Song has provided lots of examples with the Eleventh Doctor because her past is the Doctor's future.
In one case, during the "The Impossible Astronaut", she mentions "Jim, the fish," which becomes a bit of a Chekhov's Gun when the Doctor reappears later and she uses his ignorance of the incident to deduce that it's earlier in his timeline.
One highly amusing account in "A Good Man Goes to War" involves her telling Rory that she had just returned from a birthday outing with the Doctor, which involved Stevie Wonder playing at a frost fair in 1814, London, "But you must never tell him."
Less than a minute later, we get this gem:
Rory: I've come from the Doctor too. River: Yes, but at a different point in time. Rory: Unless there's two of them. River: Now, that's a whole different birthday.