Well , so we have a Doctor between Eight (McGann) and Nine (Eccleston)... should we call him "8,5th"? Or simply "The War Doctor"? Or maybe... should we call him Nine and then Eccleston-Ten, Tennant-Eleven and Smith-Twelve?
closed as primarily opinion-based by K-H-W, dmckee, phantom42, The Fallen, DVK-on-Ahch-To Nov 25 '13 at 17:44
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No. According to showrunner Moffat (in DWM), the numbering remains the same:
I’ve been really, really quite careful about the numbering of the Doctors. He’s very specific, the John Hurt Doctor, that he doesn’t take the name of the Doctor. He doesn’t call himself that. He’s the same Time Lord, the same being as the Doctors either side of him, but he’s the one who says, ‘I’m not the Doctor.’ So the Eleventh Doctor is still the Eleventh Doctor, the Tenth Doctor is still the Tenth
If you read the whole piece, he goes on to talk about how he feels the numbering works in more detail.
The problem with Tony Meyer's answer is that what Moffat has said actually directly contradicts the events of Day of the Doctor. Spoilers ahoy:
In the scene where the three Doctors are in the presence of The Moment, 10 and 11 explicitly acknowledge that the War Doctor is The Doctor, and the War Doctor accepts that this is the case.
It's worth noting that there hasn't to my knowledge been a doctor referred to by number in the entirety of new Who (my memory of the old Who specials isn't sufficiently good to comment on those), so the numbers we use to distinguish them are probably out of universe. It's likely that the current numbering system will continue to be used for convenience's sake, but there is nonetheless a very real sense that the Hurt regeneration was the Ninth Doctor.
As has been pointed out in another question, the Second Doctor's regeneration occurs off camera, and it is therefore theoretically possible that other regenerations exist between Troughton and Pertwee, but there is no suggestion in canon which I am aware of that this is the case.
I think that we should just call him the War Doctor because it would be unfair to Christopher Eccleston, David Tennant, Matt Smith, and Peter Capaldi to change the enumeration because they had already been announced as 9, 10, 11, and 12, and to change that would be not only annoying, but confusing to the actors and the audience.