It's hard to believe that a genius like Doc has no formal education. Is there any info on what his alma-mater is and what his speciality is (like, quantum physics, or something)?
This earlier script draft indicated that he had been a renowned nuclear physicist:
MARTY Doc Brown's all right — he's just a little hung up on time. A couple of years ago, he showed up at my house and hired me to sweep out this garage of his. He pays me 50 bucks a week, gives me free beer... and gives me total access to his record collection — he's got this great old record collection.
Hard to believe he was one of the world's greatest nuclear physicists.
However, there are plenty of details in that script that don't match the finished movie, for instead of Doc being an eccentric lonely inventor in the 50s, when Marty goes to Doc's mansion in the past he finds him hosting a swanky party flanked two beautiful women, and Doc comes up with the idea for the "temporal field capacitor" when he whispers something in the ear of a woman and she hits him over the head with a beer bottle.
But there is going to be a new series of Back to the Future comics co-written by Bob Gale (who wrote the original movies with Robert Zemeckis), and filling in details of the backstory of the movies, as Gale describes in this interview:
I said, "If we're going to do this, let's go back to the characters. Let's do stories suggested by the movies, by the characters. People have always asked how did Mary and Doc first meet? That's a good story. Let's do that in the first issue. People have asked, how did Doc Brown's house actually burn down? Was it a fire insurance scam? What was that? In issue #2 we tell that story. Issue #3, people have asked, wouldn't have George and Lorraine wanted to find out what happened to Calvin Klein? Well, we deal with that in issue #3.
These are all things that are suggested by the movies, suggested by people thinking about the world of the movies, and I think that our fans are going to be very very happy with the way these things are turning out. I certainly am. These comics are exceeding my expectations. Again, the people at IDW said, "Can we say that these are canon," which is always something that makes me crazy. It's fiction. How can it be canon? I said, "Look (and we actually have a text piece in the first issue talking about this) when you have a time machine and infinite time lines everything is canon, so, yes, it's all canon and none of it's canon."
For example, the first issue’s stories will uncover the origins of how Doc Brown and Marty met each other, and cover Doc’s work on the Manhattan project during World War II.
So, even if he wasn't famous as one the world's greatest nuclear physicists, I think we can speculate that he likely at least had a Ph.D. in physics, as did other young physicists who worked on the Manhattan Project (since Doc is only supposed to be in his 30's in 1955, he would have been in his early 20's during the last years of WWII), like Richard Feynman.
According to Futurepedia (see note below), in the 1950s, Doc Brown worked as a professor of physics at Hill Valley University, suggesting physics may be his speciality. This isn't such a wild leap, as building a time machine would involve a fairly detailed knowledge of physics.
Note: whilst the article I link to doesn't provide a specific reference for this, it does refer to Hill Valley University which is apparently referred to in the Animated Series. Not having seen any of the animated series, I can't confirm this, but it may be the case that this was elaborated on in the series.