Tonight's as yet unaired episode of Doctor Who entitled "The Name of The Doctor", I expect will yet again not reveal the name of The Doctor to the audience but to Clara Oswald. What of the title of The Doctor? Assuming that The Doctor is a title which he acquired on Earth, this itself has a tendency to date him in Earth's history at least.
The word "Doctor" itself comes from the Latin "docēre" but has an agentive ending, that is a suffix appropriate to someone who does that thing. A Librarian does library work, a builder does building work, a cellist does the cello; a Doctor quite literally does docēreing and since to docēre means "to teach", a Doctor is one who teaches.
In fact the word "Doctor" as it is applied to medical practitioners seems only to be start being applied from about the 12th Century. That in itself is worth noting.
It was universities who first conferred the title of Doctor on people. Quite literally a Doctor was one who was licenced to teach. The term "licentia docendi" finally crystallised to a university teacher at the University of Paris in 1213 after a series of rule changes governing the university.
Universities are generally considered to have started with the University of Bologna in 1088 which took its name from the Latin phrase "universitas magistrorum et scholarium", or if you will, a community of teachers and scholars.
Given that Doctor Who is a time traveller, it's not inconceivable that he could have been to and in fact been resident at any number of Universities from the 11th century and onwards. Presumably he is either a physician of some sort or has studied somewhere.
In "The Hand of Fear" (broadcast in 1976 - 4th Doctor serial), he says that studied at the Prydonian Academy on Gallifrey and was in the class of '92. It is also stated in "The Armageddon Factor" (1979) that he attained the doctorate along with Drax. Drax however calls him Theta Sigma which presumably is some sort of code name, which could possibly relate to something like a fraternity house. His teachers included Borusa and Azmael - the former became a Cardinal and later the Chancellor of Gallifrey; the latter, we don't really know.
Given that The Doctor stole his TARDIS it's probably a safe bet to assume that the Doctor's Doctorate is probably in Physics because time and space are both physical quantities of the universe, which is what Physics is concerned with; though given his rouge nature, he might also only hold an honorary doctorate somewhere.
So then, The Doctor is more than likely a Ph.D in Physics but who honestly has a clue of what his major would be?