Smart People Build Robots

So your resident smart guy is already bespectacled, a master chess player, and fluent in Latin. How else can you show how brilliant he is? Why, by having him build robots of course! May be justified (said smart guy has a degree in engineering, etc.) or not (said smart guy is a teenager who just happens to, among his many other skills, be able to create from scratch a fully functional robot).

Bonus Points if the smart guy uses his robot to battle the robots of other smart guys.

Compare and Contrast: Robot Master

Examples of Smart People Build Robots include:

Anime and Manga

The Professor in Nichijou is age eight and has built a perfectly functional, sentient robot she named Nano. Outside of this achievement however, she spends about 95% of her time on typical eight year old activities like coloring books.
Rin-Rin in Sister Princess builds two robots - the first shows up in her introductory scene, and building the second drives her spotlight episode.

In Iron Man, Tony Stark is kidnapped while visiting Vietnam (Afghanistan in the film version) and ordered to help develop weapons for his captors. But instead of rigging a MacGyver style escape plan, he uses the equipment he's given to build a suit of Power Armor and break out.
Henry Pym and Reed Richards, two Omnidisciplinary Scientists who build robots in their spare time. Pym's are plot-relevant, Richards' just hang around as background detail.
The Silver Age Superman could pump out a quick robot, no problem.

Anakin Skywalker in Star Wars Episode I: The Phantom Menace is shown to have a preternatural talent for all things mechanical, in particular robots. Used not so much to show smartness per se, but rather cleverness and precociousness.
In the TV movie High School USA, the two nerds built their own robot that responded to voice commands.
Revenge of the Nerds: the Nerds have a robot they use as an occasional butler.
Robin Williams in Flubber has two.
Averted with Megamind, who stops short at mere Mo Cap Mecha instead, and a Powered Armor for his assistant Minion (who depends on it, being a limbless talking fish.)
Averted big time in Real Steel, where practically anyone with just enough smarts can build robots from scrap.
Dr. Morbius in Forbidden Planet
Both Hiro and Tadashi in Big Hero 6.
Brittney Kornblum in Xtracurricular.
Tony Stark in Iron Man still works with a couple robots he built in his teens.

In the CHERUB Series book People's Republic, Ethan is a geek who is into chess and building robotics, though given that he is a kid and it is not a Science Fiction series, these are relatively simple robots and do not incorporate things like human-like Artificial Intelligence and speech.

In an episode of The Big Bang Theory, Sheldon, Leonard, Raj, and Howard, construct a robot named M.O.N.T.E. for the Southern California Robot Fighting League Round Robin Invitational, though end up battling Barty Kripke's robot "the Krippler" in a street fight.
- Also Lampshaded: There were a few jokes about them being defensive about how nerdy they were perceived by Penny, all the while doing the most geeky thing Penny could possibly imagine.
In the Nick Verse show Zoey 101, Zoey and her friends construct a robot to battle against a group of Insufferable Geniuses' robots.
On MythBusters, Grant, who formerly built robots for Battlebots, is Lampshaded as this type of The Smart Guy by the others as a Running Gag.
Screech in Saved by the Bell had his own robot that just hung around his bedroom.
Wizards Of Waverly Place's smart guy Justin Russo builds a working robot out of household items... so he could dance with it in a "big synchronized robot event".
Riptide (TV Adventure series, not the novel): Gadgeteer Genius sometimes-helper Murray "Boz" Bozinsky has a robot he calls the RoBoz.
Urkel from Family Matters built a robot or two.
In the new cast of Degrassi there are four kids in the gifted class that build a robot for their school's competition.

"The Future Soon" by Jonathan Coulton has the speaker building a robot army among other random acts of genius. It ends badly.

Tezzeret, the Seeker is one of the cleverest people in the series, and has a penchant for building automota, golems and clockwork constructs, among other things. His cards can even turn your artifacts into robots to fight for you!

In Chrono Trigger, smart girl Lucca has the skills and knowledge necessary to repair and reprogram a robot from the distant future to be her friend and party member. Good thing the future still uses C.
Tails, especially in Tails Adventure and Sonic Chronicles.
Jennifer in Disgaea built Thursday at an early age.

Inverted with Ctrl+Alt+Del, where Ethan somehow manages to make his X-Box and Gamecube into sentient Robots.
Sean 'Dark Smoke Puncher' McNinja is the family computer nerd, and fills his free time by building robot super animals. His father does not approve unless the robots hurt someone.
In Girl Genius, the title character has a particular aptitude for steampunk robots.
Kat in Gunnerkrigg Court. Annie also does it at first, but she just pieced together parts of a disassembled robot.
Subverted in Homestuck: Equius is the resident robotics expert yet he doesn't seem to be particularly bright.

Doctor Steel, building an army of giant robots to take over the world.
More than a few characters in Fenspace can build robots even though they shouldn't know how.

In Kim Possible, all the smart guys build robots and battle them at a place called "the Robot Rumble".
Toby, Word Girl's most intelligent Arch Enemy (not that there's a lot of competition for the title) builds giant robots to carry out his Evil Plans.
The Simpsons: Homer & Bart try to build a robot for the Show Within a Show "Robot Rumble." But they can't get it to work, so Homer makes a "robot" that's just him in a metal suit. They make it to the finals, where they're matched up against 5 time returning champs Professor Frink and son. So, played straight (the Frinks) and Inverted (Bart & Homer, being dumb, can't build a robot).
- Though Homer did throw out a half finished robot, he built in another episode.
Dexter of Dexter's Laboratory does a lot of advanced robotics.
Jimmy Neutron covered this trope a couple times.