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Nobody knows the Doctor's name, because none of the writers have ever invented one for him. And they likely never will. Current show-runner Steven Moffat explained that any name they give him now would be anticlimactic, and take away from the mystery of the show without adding anything new, and many of his predecessors (and equivalents in other media, like the novel range editors) said the same thing (Moffat illustrates the ridiculousness of assigning a name to the Doctor in Doctor Who Magazine #475 where he jokingly says the Doctor's real name is Mildred). We do know that he calls himself "The Doctor", and sometimes "Theta Sigma" (an old Academy nickname, and possibly later an undercover name), and other aliases like "John Smith", but none of those are his real name.
Moffat has jokingly suggested that he knows the real name, such as at the 2012 San Diego Comic Con, but this is not to be taken seriously given he also claimed at the same event that William Hartnell told him the character's name when he was all of one year old.
In-universe, of course, the Doctor knows his name, and there may be a few others.
In the past, there were many Time Lords who knew it and, as we learned from "The Day of the Doctor" & "The Time of the Doctor", at least one surviving Time Lord on Gallifrey must still know it (otherwise, the Doctor's name couldn't be used to identify the correct universe to which to return the planet). Plus, a few of his companions knew it, but they may all be either dead or unable to remember it. (We know he told Peri Brown ["Slipback" (audio story)], but she couldn't pronounce it and could only remember the first syllable; in the novels he told Sam Jones, but she's dead; Donna likely got the knowledge while she was Doctor-Donna, but can no longer access any of those memories without her brain burning out. Clara becomes aware of it in "Journey to the Centre of the TARDIS" but loses the knowledge soon after.) It should be noted that possibly Clara actually does know his name, as she even masquerades as the Doctor briefly in "Death in Heaven".
In fact, the only people we know of that know the Doctor's name are the Doctor himself and River Song (she tells the Doctor his name in "Silence in the Library" and uses his real name to open the TARDIS in "The Name of the Doctor"). And it seems likely they're the only two, period—one of the New Series Adventures novels confirms that before he told River, nobody alive knew his name, but even if you ignore the novels, it's hard to think of anyone else who'd be likely to have that secret.
And his name is a very big secret. There have been hints going back to the classic series (e.g., Lady Peinforte in "Silver Nemesis"), but series 6 made it as blatant as possible: The First and Oldest Question, the question that's the core of the Silence's beliefs, the question that should never be answered, is "Doctor Who?" (because the answer would identify the universe to which Gallifrey belongs, allowing the planet to return, possibly re-igniting the Last Great Time War).
It may be worth noting that the Doctor accidentally says his name in "Return of the Living Dad" (novel), after he has been drugged or something. Unfortunately, author Kate Orman unsportingly doesn't tell the reader what he actually says, although the character to whom he is speaking comments that it would be difficult to address an envelope with it (a reference to long Gallifreyan names such as Romanadvoratrelundar)
"Dr. Who" Controversy Edit
One controversial extrapolation from the evidence provided is that the Doctor's name is literally "Doctor Who". It is true that the end credits for the First-Fourth Doctors and the Ninth Doctor all called the character "Dr. Who"/"Doctor Who", until Peter Davison and David Tennant respectively took over the role and asked that it be changed to "The Doctor", reflecting the chosen name the Doctor insists on using the vast majority of the time. Notably, in "The War Machines", WOTAN refers to the Doctor as "Doctor Who", but this is the only televised story to have referred to him by that name.
The comic stories are a different matter. Especially in the earlier stories, the name "Dr. Who" was used regularly to refer to the character of the Doctor (see extract on the right). While this naming convention has been abandoned in more recent years, following the convention of using the Doctor's chosen name, those early comic stories provide an in-universe precedent for the name of the character actually being "Doctor Who".
Out of universe, there is more contradictory evidence. Even among the actors to have played the Doctor, you have those such as Peter Capaldi who refer to the character as "Doctor Who" in interviews all the time, where, as mentioned earlier, others such as Davison and Tennant have been so opposed to this position that they insisted on changing the end credits to call the Doctor just "The Doctor". Production staff are also divided, with those such as Moffat insisting the Doctor's name is unknown while Verity Lambert herself used the name "Doctor Who" to refer to the Time Lord. Story titles are another point of contention, as many of the novelisations were titled in the format "Doctor Who and the...", for example "Doctor Who and the Doomsday Weapon" (novelisation) and "Doctor Who and the War Games" (novelisation). The trouble with the episode end credits also flows into the Doctor Who Annuals, with the annuals from 1971-1981 featuring the words "Starring JON PERTWEE/TOM BAKER as DR WHO", but not after the Fifth Doctor took the reins.
Away from the production of the show Doctor Who, there are also lots of instances in the media of the character being referred to as "Doctor Who" or "Dr. Who", as well as the more widely accepted "The Doctor", proving the widespread usage of the name to refer to the character, not just the show. However, as the media is not a licensed source of information about Doctor Who, we can pull no more evidence from them.
There are other problems with the theory that the Doctor's name is literally "Doctor Who". First of all, there is the Doctor's insistence that people call him call him "the Doctor", "Just the Doctor" ("Rose" et al.), and that the name "the Doctor" was chosen by him as a promise ("Never cruel or cowardly.[...] Never give up. Never give in." --"The Day of the Doctor"), which suggests that "Doctor" was not a part of his original name. Why choose and ascribe so much significance to half your own name but reject the other half? "The Doctor" is also known to be an alias ("The Mysterious Planet" etc.), but most humans would not consider using part of one's real name to be using an alias. The Doctor's real name is also said to be quite alien and hard to pronounce ("Vanderdeken's Children" (novel), "Salvation" (novel), "Illegal Alien" (novel), "Slipback" (audio story)), which "Doctor Who" is not.