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.
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.
Rob Lucci from One Piece. 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.
Nami however defiantly fits this, often having to hit some sense into her captain.
Killer. His captain, Kid, is so Ax-Crazy that Killer literally does everything, including organizing black market visits and meeting between rival pirates for possible alliances, along with acting as mediator and preventing said pirates from pushing one of Kid's many, many berserk buttons.
In Naruto, 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.
While Fullmetal Alchemist's 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.
This is effectively what Golden Boy's Kintaro Oe does, although pinning him down exactly is a bit tricky.
Saru of Seto no Hanayome becomes one of these when he starts working for Mikawa. Of course, he's still not exactly competent, but compared to Mikawa himself, Saru gets a lot done.
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.
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.
Following on from Doraemon, there's an entire subgenre of this in kiddy anime. More or less normal kid, paranormal sidekick/pet. Pokémon crosses this with Mons by having Pikachu as Satoshi/Ash's fulltime companion.
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.
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...
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.
Reborn from Katekyo 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).
Yuno from Mirai Nikki acts as a sort of sidekick to Yukiteru, and throughout the series, she does most of the killing and grunt work.
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.
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.
Tengen Toppa Gurren Lagann brings to the table Kamina, the leader of the group, to which the main character Simon is a sidekick. While Kamina is the more confident and charismatic of the two, Simon is the one who bails him out repeatedly, and definitely the more intelligent of the pair. Kamina is good enough to lampshade this by refusing to take credit for Simon's victories and repeatedly telling him what great things he's destined for. Since Simon becomes leader of Team Gurren after Kamina's death and saves first the planet, then the universe, Kamina was bang on the money.
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.
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.