Sebastian, Ciel's butler in Black Butler, does most of the dirty work for Ciel, and, as he puts it, "is one hell of a butler." Of course, he is a demon with awesome supernatural powers that serves Ciel because of a contract...
In Bleach, Nanao Ise is stuck with the job of running the 8th Division. Her captain can run the division, but he loves to drink, sleep, and skirt-chase too much. Of course, Kyoraku isn't incompetent, just lazy. We see how he is in battle. Subverted in the case of Hitsugaya and Matsumoto, who behave exactly like this, with Hitsugaya often winding up doing all the things Matsumoto should be doing - except it's Hitsugaya who's the boss, whilst Matsumoto's the lazy subordinate. Played straight however between Hitsugaya and his former captain, Isshin Shiba.
In the anime A Certain Magical Index, in the angel's fall arc, Touma, who usually has to solve everything because everyone around him is incompetent, is toyed with and ultimately forced to watch while his friend, Tsuchimikado, solves the problem.
Death Note has Ryuk. Given his motivations are to stir up some fun, and given who he gets to help him...
Canti from FLCL. Everyone thinks that the kid who goes inside the robot is the hero, but the kid doesn't remember anything about what happened (even if he takes credit for it). The Robot also does chores and rescues girls.
While Roy Mustang is obviously very competent in his own right, his general laziness when it comes to such things as paperwork causes them to fall to his incredibly efficient and loyal lieutenant, Riza Hawkeye. Roy's reputation for laziness might have even reached the level of Memetic Mutation, as it is touched on comparatively infrequently, compared to some of the other Running Gags in the series, such as Ed's shortness and Al's love for kitties.Of course, this is just what Roy wants people to think to hide his real goals.
Alphonse. Even though his brother doesn't treat him as a sidekick, he's seen this way by all the other characters. He has less credentials, of course, and needs a transmutation circle to perform alchemy (until his memories of his trip through the Gate of Truth are restored). But he is a very competent alchemist on his own and a greater fighter than his brother. He even Took a Level in Badass in Brotherhood and learned a foreign form of remote transmutation.
Yuno from Future Diary acts as a sort of sidekick to Yukiteru, and throughout the series, she does most of the killing and grunt work.
Kondou Isao is a figurehead of Shinsengumi in Gintama. His number two, Hijikata, does all the managerial work and makes most decisions in the organization. Combat wise, his number three, Okita, has the highest on-screen kill count than the rest of the organization combined. Understandably, enemies of Shinsengumi generally target Hijikata and or Okita for assassination, rather than Kondou.
This is effectively what Golden Boy's Kintaro Oe does, although pinning him down exactly is a bit tricky.
In Haruhi Suzumiya, Yuki Nagato has almost godlike powers, but acts more or less as Kyon's sidekick. She solves most of the problems created by Haruhi and generally acts as the Brigades' personal Deus ex Machina. The unofficial leader of the team claims to be the only one of them without any special powers (According to Itsuki, Kyon is completely normal, and his Agency have done extensive background checks that prove it), but there is something unusual about Kyon; at the least it's being "god's" favourite.
Subverted every which way in Kamen no Maid Guy. Main character Naeka is as useless as the stereotype and her maid/bodyguard Kogarashi is powerful enough to solve any problem she might have, but the Maid Guy's bizarre methods have everyone else trying to stop him from actually doing anything.
Reborn from Katekyō Hitman Reborn! acts as a "tutor" and sidekick to Tsuna. In much of the beginning of the series, most of Tsuna's problems are solved by Reborn. In fact, two of Tsuna's guardians, Hibari and Mukuro, are much more capable than Tsuna (though of course, it's later suggested that he surpasses them in the future).
The Daikenja in Kyou Kara Maou without whom Shinou would have been defeated by the enemy he was fighting against. This is explicitly stated. While Shinou is more powerful in terms of both magic and physical strength it was Daikenja's strategies and nearly preternatural foresight that allowed their army to secure a win over the Sovereign.
Subverted in Legend of Galactic Heroes. While Reinhard von Lohengramm is a genius in quite a lot of things, he recruits people who are better than him in certain areas (Kircheis, Reuenthal, and Mittermeyer in the military; Mariendorf and Oberstein in politics and governing) in order to be truly great.
Sun has two in The Legend of Sun Knight. The first is Storm, who holds a similar rank, but must still take orders from Sun. Generally, work that Sun doesn't get done will be pushed off onto him, and as the one everyone goes to for gossip, Sun uses him to gather information. The other is Adair, one of the most loyal knights in Sun's squadron. He does roughly 3/5 of Sun's work on a regular basis and can actually interpret the intent behind Sun's praise laden speech (recognizing a lecture on how the God of Light is merciful to secretly be an order to put a bag over someone's head before beating them up.) One several occasions it's mentioned that the squadron under Sun can actually run just fine in his absence, since Adair handles so many of his duties anyway.
Neuro from Majin Tantei Nougami Neuro decides to "work for" protagonist Yako in her detective agency (that he forced her to set up). At least on the outside. In actuality, he's the one solving all the mysteries and is actually a sadistic Daemon from another world.
Kabuto is this to Orochimaru. Kabuto isn't as strong (or manipulative) as his boss, but he is much better at making plans, and his medical skills have greatly helped his master's unethical experiments. Orochimaru says he doesn't know what he would do without him.
For much the same reasons Shikamaru is this to Asuma. On one particular occasion Shikamaru was put in a team with people that were all 10 years older then him at least, and they were soon remarking how they would have been wiped out very quickly without him.
Kakuzu is something of this to Hidan. They're partners and thus of equal standing, but while Kakuzu is a calculating, forward-thinking, highly competent tactician who tries to accomplish goals in the most pragmatic way possible, Hidan is a psychopathic, blood-crazed moron who relies on his immortality to compensate for his complete disregard for caution and inability to even acknowledge his limits, but he's not invincible and thus very easy to incapacitate enough to take him out of the fight. Without Kakuzu there to think for him and rescue him from his bad decisions, he would have been taken out a long, long time ago.
Mendoza in The Mysterious Cities of Gold: He's easily the most competent fighter in season 1, holding his own against entire squads of Olmecs and Spaniards, and rarely takes an injury. He's also the only level-headed adult in the group (which isn't hard when you consider Sancho and Pancho are dumber than rocks), and both a famously skilled navigator, knowledgeable about survival, a cunning strategist and quick thinker. He's half the reasons the three kids survive season 1. As part of his character development he seems to become aware of his sidekick status, becoming far more invested in helping the children and protecting them than his own goal of becoming rich. Interestingly, he also shows a good example of this trope interacting with Competence Zone: He's awful at solving the various riddles and traps left by the Precursors, which leaves the three protagonist, Esteban, Zia and Tao, to do the plot critical stuff, while Mendoza focuses on keeping them safe.
Omamori Himari has Yuuto, an heir to an demon-slayer family who's not good for much of anything. Himari shows up to protect him until he learns what he's doing, but thirty chapters in so far it looks like his only talent is building demon harems.
Zoro's foil Killer: arranging all his captain Eustass Kid's black market deals and meetings and generally making sure his captain doesn't blow them due to his temper.
Nami however defiantly fits this, often having to hit some sense into her captain.
Rob Lucci Not only is he far, far, far (etc.) stronger than Spandam, but by being a General Failure, Spandam shouldn't be a boss at all. It seems, that Lucci doesn't try to take over the leading chair only because unquestioning loyalty to his superiors was hammered into him from childhood.
In Ouran High School Host Club, one gets the impression that the Club wouldn't survive without Kyouya keeping an eye on Tamaki. He decided early on after they met that he would do everything in his vast intellect to make Tamaki seem as awesome as possible for Tamaki's sake, even at the cost of his own reputation. Every club member knows this and calls Kyouya the Shadow King. Tamaki even spends an entire event trying to get Kyouya to compete against him. This has a rather depressing explanation: As the youngest son, Kyouya is encouraged to do well but never deliberately outshine his siblings; being a Hypercompetent Sidekick was his only outlet. Subverted in episode 26 of the Anime, when Kyoya's father claims it was all a Batman Gambit to get Kyouya to excel at something he cared about, though given the circumstances of his confession (Kyouya engineered a hostile takeover of his father's company just to prove he could) he may just be trying to save face.
Minako Aino/Sailor Venus to Sailor Moon, with Minako being easily the most battle competent and experienced of the Inner Senshi-something all of the Sailor Sensi know well. In fact, some manga side stories imply the two sometimes go superheroing around Minato ward with the role of the sidekick taken by Sailor Moon...
Tengen Toppa Gurren Lagann brings to the table Kamina, the leader of the group, to which the main character Simon is viewed as his sidekick (though Kamina would insist they were partners). While Kamina is the brash, confident and charismatic of the two, Simon is the one who actually gets things done and definitely the more intelligent of the pair. Kamina is fully aware of this of this dynamic between the two and he will lampshade this often. He fully acknowledges Simon being the true power of their duo (and of Team Gurren in general) and will often do his best to help Simon believe in hismelf and for their comrades to believe in him (though outside of a couple few, most didn't really let this sink in.) Kamina is proven right beyond all expectations. After Kamina's death and Nia is able to help Simon believe in himself and move past the pain, he takes the leadership and leads the now believing Team Dai-Gurren to overthrow Lordgenome and years later, defeating the universal rulers, the Anti-Spiral and saving the universe.
In The World God Only Knows, Keima is this to Elsie. He's so good at capturing runaway spirits that he actually managed to win an award for Elsie which gets taken away because the higher-ups couldn't reconcile her outstanding record with the klutz and not-devil-like Elsie at all. This trope is lampshaded during the Shiori arc, wherein he states that partners are nowadays supposed to be intelligent and competent, and Elsie remarks that Keima is the perfect partner by his own definition. He retorts by saying she is supposed to be the partner.
Many teams in World Trigger have hypercompetent aces and weak captains. In particular, Tamakoma 2 and Kuruma Unit have highly ranked aces that are many times stronger than their captains.
Rishid in Yu-Gi-Oh! acts as The Dragon, but if one looks at his record, he seems to have a definite edge on his boss (and brother), Marik. Marik lost to Yugi multiple times through proxies during Battle City, despite regularly cheating or otherwise playing dirty. Rishid, meanwhile, won twelve Locater Cards over the course of the tournament (enough for both his boss and himself to qualify), and the only time we see him Duel, he pretty much beats Jounouchi around the block while playing completely fairly. However, Jounouchi realizes that Rishid was probably not Marik because of he's playing fair, forcing Marik to have Rishid play a fake Winged Dragon of Ra to maintain their cover (since Marik was posing as a new friend to them). Unfortunately, Ra did not like that at all and unleashed its wrath, with Jounouchi winning because he was the first to recover. (Conversely, Dark Marik would have lost to Jounouchi if not for a technicality (namely, Jounouchi collapsing due to having endured an attack from Ra's Phoenix mode in the Shadow game (thus, feeling every bit of pain).)