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 also said that the creator of the show gave Moffat the Doctor's name. After revealing this he said that none of the actors knew the Doctor's name. So out of these two answers you can conclude one thing: Moffat is not a reliable source. Matt Smith could be though. He said that Moffat does know the Doctor's name and wouldn't tell him. if you want the link to this interview it is https://www.youtube.com/watch?v=Nt-bV6OQMRM
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, 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)