Dr. Isaac Gilmore from Cyborg 009. Also of the "ex Mad Scientist" type, one of the most memorable episodes of the 2001 series depicts with very brutal details how Gilmore, with his reputation in tatters after a failed experiment, is taken in by Black Ghost and later defects, unable to deal with their horrible methods anymore. There are also Dr. Finder, Dr. Ross and Dr. Kouzumi...
The Beast fits this role even better. Despite being a Doctor...
Forge also had elements of this while on the team.
Professor Calculus from The Adventures of Tintin, invents some remarkable devices ranging from a submarine to a sonic weapon, works on a Moon rocket, somewhat eccentric, quite deaf.
The Count in Spirou and Fantasio can extract almost any chemical from mushrooms, build a submarine and electronical mind-control countermeasures, is old fashioned in his way of dressing and speaking, and can be very absent minded at times.
Adhemar from The Adventures of Nero, who looks like a baby in a smoking.
North By Northwest has a character who is literally referred to as "The Professor," although he's less of an impossibly brilliant man of science! and more of a cloak-and-dagger spy type.
Henry Walton "Indiana" Jones, Jr. Ph.D. from the "Indiana Jones" movie series is also a professor of archeology when he is not fighting nazis, thuggees or recently communists.
Leonard of Quirm, in the Discworld novels. An exaggeration of Leonardo da Vinci, Leonard sometimes appears to be a one-man Industrial Revolution.
Partially subverted in that Lord Vetinari keeps him locked up in a prison (though Leonard is having so much fun inventing stuff he doesn't notice) so he can't throw off the balance of the city or give gadgets to the heroes.
On the other hand, it's pretty hard to imprison a man who lives mostly in his own head, forgets to eat and often doesn't see how people could to turn his Moving Mountains Out of the Way Device (for mining, obviously) into something more sinister. Leonard of Quirm has all the cunning of a concussed duckling, and Vetinari keeps him supplied with, well, everything, and tucked safely away behind booby traps he himself designed.
Abraham Van Helsing in Dracula. Is there nothing he doesn't know about? Could also be seen as a forerunner for the idea of the genius with the funny foreign accent (he's Dutch) as well as a medley of eccentricities.
Several novels by Jules Verne, especially The Mysterious Island.
Faust is possibly the granddaddy of this trope. Although he has been presented in different lights, Christopher Marlowe depicted him as the most learned scholar on earth but so hungry for more knowledge that he sold his soul for it. Once this happens, instead of doing anything intelligent, he hits the pope with a fish. It makes sense in context.
Blackberry of Watership Down, given his limitations as a rabbit, is also The Professor. He may not understand the physics involved, but he'll devise plans that rely on buoyancy, tensile strength or leverage just the same.
Live Action TV
Marco Pacella (and, later and to a greater degree, Dr Kevin Burkhoff) from The 4400.
Walter White of Breaking Bad exemplifies this trope. From breaking into a high security chemical storage facility using the thermite from an etch-a-sketch to using mercury fulminate in order to blow up an entire floor of a drug dealer's headquarters, the man's chief weapon is his degree in chemistry.
On Doctor Who, Ace called the Seventh Doctor "Professor". Despite his irritation at this sobriquet, all of the Doctor's incarnations have had at least an element of The Professor about them.
And, of course, there's Professor Yana from Utopia. However, he cheerfully admits it's an invented title, since there hasn't been an university around in centuries.
On the Spin of of Doctor Who. The Sarah Jane adventures episode "Death of the Doctor" says that Ian and Barbara became Professor. note Okay, it was also said they hadn't aged since the 60's, though that could mean anything.
Sayid is LOST's Professor, creating and fixing all sorts of gizmos using only what he can find in the plane wreckage or the hatch. Of course, none of these gadgets got him off the island though he did indeed get home in season 4.
Until they introduce Daniel Faraday who is, in fact, a theoretical physicist.
On The Next Generation, Geordi Laforge and Data often trade off on this role. Whenever the crew needs to do something that doesn't involve diplomacy or combat, it's good odds one (or both) of them will develop the Applied Phlebotinum solution.
Daedalus from Greek Myth makes this Older Than Dirt, having build the Labyrinth of the Minotaur and pairs of wings that allowed him to fly, among many other things.
In his footsteps follows Archimedes, a Greek inventor from the Greek colony Syracuse in southern Italy. He is best known for his work on lever action and bouyancy. While it's not entirely clear what kind of devices he actually invented and how well they worked, but he had a massive reputation as an inventor even during his lifetime.
Imhotep was a genius who lived in Egypt 2600 BC, credited with strongly contributing to the founding of the sciences of engineering, medicine, and architecture. His service to the Pharaohs was so remarkable that he became the only commoner ever to be elevated to godly status after his death.
Half-Life is full of these. The first game has them as allies who will open security doors for you (portrayed with three or four generic character models) while the sequels all have specific characters, most notably Dr. Kleiner who fits the archetype to a tee. Dr. Vance appears to be the team leader, though he comes of more like a Cool Old Black Guy. Dr. Freeman appears to be more of a lab technician, but he is a theoretical physicist working on teleportation as well.
Elco from the Sleeper HitWestern RPGSudeki pretty much nails this on the head and drills it home with magical laser guns. He does actively participate in a lot of fighting, though, so he's a bit different from the standard all-support role.
Roughly half the cast of the Metal Gear games qualify for the trope in at least a few respects, although Otacon probably comes closest to fulfilling it.
Grant Florian of Magical Girl Lyrical Nanoha As Portable: The Gears of Destiny, the father and creator of the Florian sisters. A brilliant and benevolent, if sometimes overenthusiastic, scientist who devotes most of his time trying to find a way to reverse the death of his home planet.
Horace from The Last Story functions like this. He proclaims himself an archaeologist, but he seems to know a lot about architecture, geology, and dark magic (merely as an academic curiosity, of course), aiding the protagonists in a series of optional chapters.
Riff from Sluggy Freelance, despite being a master of wacky hijinks and Brilliant, but Lazy, is actually a very skilled scientist who invents anything from a Dimensional Flux Agitator to a toaster that can shoot toast through walls.
Dr. Amos Messing of the Amazing Three, in the Whateley Universe. Not that he's a parody of anyone else on this list whose initials might be 'RR'...
A minor character in Darwin's Soldiers: Card of Ten is referred to only as "the professor."
The Global Guardians PBEM Universe has both Herr Doktor Archeville and The Academician. The former is a world-renowned scientist who uses his advanced technology to fight crime. The latter is the leader of the People's Republic of China's official hero team. Both are among the smartest men in the world.
Professor Membrane Invader Zim is a prime example going so far as to make the Perpetual Energy Generator AKA P.E.G.
"The man without whom this world falls into chaos, and the inventor of SUPER-TOAST!"
Professor Utonium from Power Puff Girls is possibly one of the kindest and most fatherly characters here- although in one episode he realises that pretty much everything good he ever invented was by accident, from the eponymous girls to Sloppy Joe's.
Also a parody of the original Jerry Lewis version of The Nutty Professor.
Professor Hubert Farnsworth from Futurama is a subversion. He's capable of making many wondrous and useful inventions at times and some incredibly odd/hilariously simple ones but due to his age (160 years old as of the second season), he often has many odd habits that belie his intelligence. He also has Mad Scientist tendencies, perhaps more so than Doc Brown from Back to the Future.
Dr. Benton Quest from Jonny Quest is an especially egregious example. One must presume he holds multiple degrees and doctorates, as he is apparently qualified to work in an absurdly diverse number of science fields from anthropology to astrophysics without much assistance.
Parodied in the form of Dr. Thaddeus Venture of The Venture Bros., who is not actually a doctor, having dropped out of school two months before getting his degree. While most of his inventions do actually work, his success is often due more to blind luck and other people's perceptions of him than any actual skill. Also an example of a Mad Scientist, as many of his inventions are monstrously destructive or unethical, including an "Ooo-ray" that somehow melts buildings with sound waves, and a dream machine powered by the heart of an orphan. Venture's twin brother, Jonas Venture Jr., his good fraternal twin, is a more straight application of the trope, but is still played for comedic value.
In the sequel-series Extreme Ghostbusters, Egon Spengler found himself playing his role to the new blood.
Cornelius Robinson, the older form of the protagonist in Meet the Robinsons. Graduated college at age 14 and went on to invent flying cars, self constructing buildings, time machines and much of the rest of the future's technology.
King of the Hill gets a quick shot-in at this when Dale believes he's been transported into a distant future. As he looks around the landscape, he remarks, "It's times like this I wish I had a friend called 'The Professor!'"