(SPOILER)While the most popular theory about John Hurt's Doctor seems to be that he was from the Time War, people on TV Tropes are also theorising that John Hurt may have been an incarnation from before William Hartnell's Doctor:
Other theories:Assuming he actually is a previous incarnation, then there are only two possible points he could have come from; between the Eighth and Ninth Doctors, or before the First.
Consider what was said about the secret, "you've been running from your whole life". The Doctor who started running was the First. So the secret dates back to him. So, although he was the first Doctor, he was not the first incarnation of that Time Lord. He changed his name after regenerating and started running from whatever he had done. Yes, the First Doctor wasn't really a good guy when we met him, but he was still reforming. Perhaps at that point he was thinking that he'd already done the worst he could, and couldn't really be any worse.
This would be consistent with the multiple pre-Hartnell Doctors implied by The Brain of Morbius.
The EU toys with the idea that the Doctor was, in an earlier life (via either an incarnation before the First, a literal reincarnation, literally made from his raw material, etc), a Time Lord known as "the Other", who co-founded Time Lord society alongside Rassilon and Omega.John Hurt's Doctor is the one who killed the Time Lords, but also the one who made it necessary to kill them. During the Time War, the Eighth Doctor found out that he will one day destroy the Time Lords. He couldn't do anything about it, but then he died and regenerated into much darker John Hurt's Doctor. This Doctor decided to save the Time Lords any way he could, and came up with a plan, the same plan seen in The End of Time. However, he then realized he didn't want to be responsible for that, and decided to stop it only way he could: by killing all the Time Lords. He then went to Trenzalore to await his death and regeneration into Nine. So the other Doctors hate him, not because he killed the Time Lords, but because he made it necessary to kill them.John Hurt's Doctor is exactly what the Doctor said he is. He's the incarnation between 8 and 9 who did such utterly terrible things during the time war that he does not consider himself the Doctor. This would make the Ninth Doctor the 10th incarnation, but not the 10th Doctor, and so on. This means Simeon's mention of the Valeyard may become relevant in the near future, as that would make 11 the twelfth incarnation.
- Perhaps John Hurt's Doctor is from a time BEFORE he took on the persona of the Doctor. Before becoming the 1st Doctor, he acted closer to the Time Lord Victorious, which led to a event that broke him so greatly he ran.
- The way 11 spoke to him and the way he replied sort of implied that it was an incarnation that had been the Doctor prior.
- He is the incarnation between Eight and Nine, as we have never seen Eight regenerate and this is the only possible place he can be from because the Doctor clearly remembers him as being in the past. However, given the Doctor's free admission of committing genocide and doing other horrible things to end the time war, the only explanation is that this one did something much, much worse to the point that The Doctor wants to expunge him from all memory for all time forever.
- We never see Two regenerating into Three either…
The John Hurt's Doctor is not the past or future Doctor. He is not the one who stopped the Time War. Nor did he end some nebulous future conflict. He is the Doctor from the poisoned timeline created by the Great Intelligence whose every victory was turned into defeat, who gave up the moniker of the Doctor, and who perhaps destroyed his entire universe in the name of peace. In other words, he's the Valeyard.
- Normally, Valeyard guesses annoy me, because he's never been mentioned once in the new series... until this episode, when the Great Intelligence listed him with other names the Doctor would have... so if this isn't true, Moffat has deliberately tried to mislead us, because there is no way he couldn't realise how people would read into that.
- If he's not a previous Doctor (Which This Troper really hopes he's not) he has to be the Valeyard. After all, the Valeyard was the Doctor, albeit a future one. And this Season introduced the Great Intelligence as played by a different actor why couldn't the Valeyard?
- He can't be one of the eleven we know, as Clara explicitly set him apart from them (and called Smith the Eleventh Doctor). On top of that, the Doctor's dialogue suggests he already knows who this Doctor is, which means he may not be the Valeyard since that's a pre-Twelfth incarnation.
- The Doctor already knows who the Valeyard is. On the other hand, the Valeyard is not the kind of person to do things "in the name of peace and sanity".
- You never know. He might have had Character Development over time. Or he could simply use the claim as justification for his actions; plenty of other Doctor's foes have claimed to act for greater good while gratifying their selfish impulses.
I'm not sure which of these I like the most. I do, however, like the theory that Hurt's Doctor was what Eleven saw behind the door in "The God Complex", which seems quite likely now.John Hurt's Doctor is no one other than the Valeyard
The valeyard is all what is bad about the Doctor, and the Doctor is dead afraid of both of them. With time being rewritten, there is no need to use the old actor
John Hurt's Doctor can't be the Valeyard, the Valeyard is all the hate and malice the Doctor didn't express. John Hurt's Doctor did commit a terrible act, but still acted with some form a conscience.
- You only have his own word for it. Just because he claims that his actions are justifiable doesn't make it true. The Cybermen, for example, always insist that they are working in the name of peace and equality. Also, while Valeyard is all the hate and malice that the Doctor has pent up, he is still the same person and is likely to have a conscience, just monstrously warped and capable of justifying almost any action in the name of greater good.
- The Valeyard does not have a conscience, in every appearance he had in the classic series he was defined by his selfish desire for self-preservation and egotism, what John Hurt's Doctor did, the Doctor accepts, but not as the Doctor.
- The new series has added extra depth to characters like the Master. Giving Valeyard similar treatment would fit the trend. As I already mentioned, he is still the Doctor and has his characteristics, albeit horribly warped.
- Furthermore, even the Valeyard could, for example, save the universe if his own life was at stake, even if he didn't have a better reason for it.
ETA: This Slate piece is really good. Massive spoilers, sweetie. This piece on Tor is good too.