A truly astounding example: Dr. McNinja. He's a doctor, and a ninja (and constantly masked, to boot). That alone should have been enough, as each profession is the bunny ears for the other. But he's also obsessed with Batman, plus everything else about him.
Dr. McNinja is self-employed. Judy, on the other hand, is a good receptionist who happens to be a gorilla.
Danny Hua in Nukees is a sweet-tempered Cloudcuckoolander who can't follow most basic metaphors, is compulsively honest and has a phobia of speaking with contractions. He is also a brilliant nuclear engineer and one of the finest weapon designers in the world, being personally responsible for the creation of the Giant Robot Ant. There currently is a small contingent of groupies who have decided he is the Wise One for his incomprehensible, unintentionally profound-seeming speech quirks. He is also, according to author Darren Bleuel, based on a real friend of his, and there is a page full of quotes to prove it!
What about King Luca? He literally believes himself to be royalty, wears a crown and cloak, and routinely grants land, nobility, or knighthood to his followers. And he does have followers, because he's a wonderful physics tutor who singlehandedly helped many students pass their classes.
And Lord Shojo, who pretends to be senile in order to ward off assassination attempts, but was able to fool not only the paladins he controls, but also the entire Order of the Stick, despite not having any levels in a PC class.
In Sluggy Freelance recruiting Bunny Ears Mad Scientists (but keeping them at a safe distance) is basically the whole point of the Hereti Corp freelancer program.
Doug from Fletcher Apts. In high school, he was head of the Science Honor Society and #2 player in the chess club. In college, he's the vice president of the psychology club. Also, he's known for barfing easily and freaking out over female breasts.
Taisei from Sakana is a very goodhearted, yet silly and naive person who gets drunk often and is a walking source of Ho Yay. He's also a successful assistent director for a pretty big company and the smartest man of the whole cast.
Rayne Summers from Least I Could Do. Marcy McKean - Rayne's boss and CEO of the company - puts up with his Red Bull addiction, childlike behaviour, and non-existant attention span because he is genuinely good at his job.
Othar is a "Spark", and Sparks are the incarnation of this trope and Mad Scientist turned Up to Eleven. They're all highly competent geniuses and supergeniuses; and are all completely and totally barking mad.
Collar 6: Sixx is the very successful owner of a chain of hotels, so no one really cares that she has a harem of hundreds of (willing) lesbian slave girls.
But, considering the background material on the world the author has recently been providing, this is not considered to be in any way unusual ... in fact, it's considered rather admirable.
Blade Bunny is a bunny ears assassin. A lot of people don't take her seriously because of her style, but then they see her in action.
Darkly subverted in Jack. Doctor Thalmus thinks that developing a cure for cancer makes his crime of pedophilia acceptable, even going so far as to hold the cure hostage when Aurthor finds out. No one else seems to agree with his viewpoint.
MSF High. In an example that counteracts Doctor McNinja...the entire staff. The sanest people are the Mad Scientist who is addicted to coffee, and the Vampire with a harem. Name a staff member who is not certifiably insane, and frighteningly competant.
Sarabeth Phoenix of Wright as Rayne is the fairly serious head of both a company and a criminal group, but is big on anime and dyes her hair blue.
Gatemaster of Survivor: Fan Characters is a Cloudcuckoolander who considers his alliance a group of 'friends', and is perhaps one of the most competent strategists in the game, managing the orchestrate (through a combination of blind luck and coincidence), the final four of his choosing. And he does this without even realising he is in charge.
Homestuck: oddly enough, the many variants of the Courtyard Droll, who is completely unprofessional, willingly dresses in the most ridiculous outfits possible (enormous and bizarre hat completely mandatory), has a silly happy dance that he pulls out at the slightest opportunity, and hatches absurd plans involving things like wearing C4 on his head, smuggling a pumpkin full of knives into a prison cell, or constructing a bomb out of shaving cream...which work. Next to his boss, he was one of the quickest members of his gang to actually rack up a body-count.
A number of the characters in Grrl Power fall under this, from Math (lecherous but unspeakably talented martial artist), to Heatwave (capable of sublimating tanks, but dumb as a brick), to Sydney (has a suite of eight orbs that give her unique and strong powers along with comic book knowledge to apply those powers properly, but severe ADHD, trouble with authority, and overall a PR nightmare). This is explicitly justified; powers are rare enough that they can't pick and choose their team. As long as a super doesn't have an actual criminal record (and possibly sometimes even then), they're invited to join, since it's better to have them on the team where they can be monitored and trained than off it where they can get into trouble.
Apparently, this is a requirement to be a demoman in Tagon's Toughs in Schlock Mercenary. Corporal "Pronto" Pontucci had rare moments of weirdness, but was basically intelligent, competent, and most of all, relatively sensible, aside from an odd tendency to use Duct Tape for Everything. His replacement, Lieutenant Shore "Pi" Pibald, is nuttier than a pistachio plantation and several orders of magnitude more eccentric than Pronto ever was, on account of needing significant amounts of medication to come down to the level of calling himself "God Emperor Pi the First." Pi might be every bit as irrational as his nickname suggests, what with being a paranoid schizophrenic megalomaniac, but he demonstrates a level of explosives knowledge and technique second only to Kevyn, and he has Kevyn beat for pure Mad Bomber tendencies—he's thus far the only person shown to have personally blown a hole through a battleplate through a simple combination of having both an excessive amount of explosives and no self preservation instinct whatsoever.
Mikkel, The Medic, has a dark deadpan sense of humor and a tendency to mercilessly troll everyone. However, he is also all-round competent, taking care of unglamorous but necessary stuff like decontamination, heavy lifting and cooking and is generally a much needed emotional anchor.
Emil, the Cleanser, is greener than grass in may, a textbook Spoiled Brat and has the worst case of Foot In Mouth Disease since someone put a microphone in front of Prince Philip, but he knows his booms and from what we've seen so far his instincts and training are solid, and several of his interactions with Lalli and various kids show him to be quite a softie.
Lalli, the Scout, is just plain weird. Basically, he acts as much as a cat as is possible without growing a tail, has huge problems with verbal communication and tends to sleep in strange places. He is also an extremely competent scout, capable of finding his way in unknown territory with no apparent effort, very good at reading non-verbal communication (in fact being able to have several meaningful conversations with Emil, in spite of them not having a word of any language in common) and a mage of frightening power. So far, he has shown himself to be able to spirit-walk, control the weather to some extent and combat hostile spirits in the waking world.
In Darths & Droids, Darth Vader has a bad habit of executing every underling who makes the slightest mistake. Yet Admiral Firmus Piett manages to survive and continue to be a very valuable officer, despite his tendency to make If My Calculations Are Correct and Captain Obvious observations and sometimes sassing directly to Vader's face. Vader acknowledges that Piett tends to walk the edge a lot.
Vader: Your competence and your impertinence continue to maintain an unsteady equilibrium.
Piett: Thank you, Lord Facemask.
In Bob and George it would be easier to list the characters that aren't this trope. As it stands:
First off, Megaman himself. Before Doctor Wily betrayed Doctor Light, he sabotaged Megaman's programing to make him an idiot so that Light would not have anything to face his Robot Masters; Doctor Light did his best to program him to fight the robot masters, but could only re-prioritize "To be an idiot" as his secondary mission. He is frightfully-effective at stopping Doctor Wily, but outside of that he's a huge moron obsessed with ice cream and drinking beer until he pukes - in spite of being a robot.
Doctor Wily is one of the greatest experts on robotics in the world, his genius is only outmatched by his ego and insanity; robot masters are capabable of taking out entire armies with an arm tied behind their backs - each - and yet every last one of them has a... strange quirk that sends them straight into this trope. He also made Bass to be greater than Megaman in all aspects; Bass is stronger, faster, and an even bigger idiot.