13, and he's on his 11th now.
This limit has been heavily implied to not be the result of nature but rather Time Lord Law. The Master, for example, had already used up all his regenerations but tried to give himself a new set at various points in the series (at at various points did indeed receive additional regenerations from the Time Lord government). I'm not sure what incarnation the Master is on, but he has regenerated at least 15 times, I think.
Also, to note, they never said that the First Doctor was on his first incarnation. And when the 10th Doctor was wounded and produced the Hand-Doctor (in The Stolen Earth/Journey's End), did that use up a regeneration? Suffice it to say, there are enough loopholes that the Doctor can reasonably keep coming back as often as the series needs him too.
The Master's been continuously getting new bodies (or getting resurrected), though, so it's hard to say. I don't think the Time Lord law thing is correct, it just... seems to be their natural limit. It'd make a nice dilemma for when they do reach that point in the series and the Doctor needs to find his loophole.
Nah, the First Doctor is the First Doctor, since the Eighth Doctor referred to himself as Eight in the TV movie, and One calls himself "the original" in The Five Doctors. There's a lot of novels and audios that support this too. Oh, and although Dimensions in Time isn't canon (thank fuck), Susan refers to her grandfather as "my Doctor, the original". As for the Hand-Doctor, who knows? I hope it didn't use up a regeneration, 'cause that's something I still feel kinda ill about.
