This question addresses why it's a bad idea to try to do a lot of independent work before PhD in the hope of finishing PhD quickly.
Nevertheless, are there famous examples of people who have done so much work before PhD that they get a PhD pretty much automatically (say, 1-2 years or less)?
The closest example I can think of is Mihai Pătraşcu. After publishing several extremely strong results as an undergraduate, Mihai earned an MS in one year and then a PhD in one year, all in computer science at MIT.
Ludwig Wittgenstein was awarded his PhD from Cambridge in June 1929, having enrolled as a student in January of the same year. The PhD thesis was an English translation of his Tractatus Logico-Philosophicus, which had been published in German in 1921 (under the title Logisch-philosophische Abhandlung).
Wittgenstein circumvented the usual rule on terms of residence, because he had previously been resident as an undergraduate student in 1911-1913; he left Cambridge in the summer of 1913 without proceeding to a degree. Further, the reason for his being awarded the PhD at all was in order to make him formally eligible to teach at the university: there was no doubt as to his abilities.
This happens, not often from super-gifted young researchers (although such do exist as outlined in other answers) but from people who are already professional researchers but do not have doctorates.
Some universities in the UK allow people like this to obtain a "PhD by publication", where a body of existing work - perhaps 3-5 high quality papers - is linked together by a (sometimes lengthy) narrative and submitted. Typically a doctorate by this route must be completed in one year rather than three. (some universities also now offer a three year "by publication" route for new researchers, but that has a different intent)
George Dantzig solved two previously unsolved statistics problems while in graduate school (without realizing it). His advisor told him that he would accept his papers as his thesis (though I'm unable to confirm that he actually did this).
In several countries in Europe (I can confirm by personal experience Spain, Austria and Sweden) there is the concept of "PhD by papers".
This means that the PhD Dissertation is a set of papers put together with a unified introduction (motivation, state of the art). Although some of these papers can be extended, they are pretty much the published version.
If you already have the papers, the whole process can take as little as 3 months.
Also I can't help to mention Honoris Causa doctorates, which are awarded by universities to people who have accomplished outstanding contributions to some field of knowledge. These are the only ones who are awarded "automatically" to somebody.
There are several types of PhD - one which is, in effect, a book, one which is, in effect, a series of papers, and one which is, in effect, a single giant paper.
The first form is common in arts and social sciences here in Ireland, and many of theses end up being published as books.
The third format was the usual format here, both in physical and biological sciences, and was often laid out as 'Introduction', 'Materials and Methods', 'Results' (usually more than one chapter), and 'Discussion'. This is falling out of favour, because it is too hard to produce papers from it.
We are moving, slowly, to the second form, which is the norm in the Netherlands, Sweden, and Finland ( and may be in other places too). This is a short introduction, a series of papers, or chapters in paper format, some of which have been published, possibly a linking narrative between chapters, and a final discussion section.
I've supervised one person, whose PhD was ten papers, eight published, and two or so, in draft. He was (and is) a capable and prolific researcher, who had just never got around to a PhD, and was applying for a chair. Needless to add, he got the PhD and the chair :-)
Apparently Lars Onsager (Nobel Prize in Chemistry, 1968) did not have a PhD when he was admitted to the faculty of Yale's Chemistry Department. Thus he was informed that he could merely submit one of his previous publications, as a formality, in order to satisfy the requirements of a doctorate. However Onsager did some original research anyway on Mathieu functions and was subsequently admitted to the doctorate.
My father got his PhD in Theoretical Physics instead of a diploma in Munich: they counted his diploma thesis as a PhD thesis and his defense was accordingly quite longer and with a different setup (and the relatives waiting outside the examination room were rather worried at first because he took so much longer than anybody else and then floored as he had not bothered to clue anybody in in case it did not work out). However, this must have been about 1964 or so and the respective examination regulations do no longer permit this kind of shortcut and have not done so for a long time.
So this is not more than a historical anecdote.
Schmitt found the Schmitt trigger circuit as a part of a project in his bachelors. He got a PhD for that.
