Sora from Kingdom Hearts has the honor of being the sidekick to most of the Disney Universe. In any given world, his first instinct is to help that world's main character with whatever's troubling them. In most cutscenes and story events, that character will take center stage (since the worlds are adaptations of Disney Movies). However, Sora is far more powerful than any world companion and is always the one who takes down the main villain for each world (this is in fact a gameplay element—a boss can only be killed by Sora's attacks.). He's often also the one who provides the initiative for said main character to do whatever it is he needs to do.
One of the Horde starting zones of Wrath of the Lich King has Varok Saurfang, grizzled veteran and Memetic Badass of the game, working in the shadows to mitigate the effects of the reckless commands of Garrosh Hellscream, a young and brutal but successful commander, whom he fears might reawaken the dark side of the orcish race.
In Cataclysm (and for much of Mists of Pandaria), Saurfang becomes a Shell-Shocked Veteran and, in his place, the player becomes a Hypercompetent Sidekick to Garrosh, who was made the defacto leader of the Horde. As Garrosh wasn't any less reckless in Cataclysm/Mists than he was in the previous expansion, the player often found themselves cleaning up his messes selflessly.
In fact, in a lot of quests, scenarios, and raids in the game, the Player Characters often take up this role. Sure, sometimes the NPC heroes may show up and swing their hammers around (looking at you, Arthas in the 'Culling of Stratholme'), but when push comes to shove, it is often the coordination and skills of the players themselves that really gets things done.
Drakuru is one of these to the Lich King, while at the same time you play this role to him. When you learn who his allegiance is with though, you turn into The Mole and sabotage his plans.
Flea and Slash are far more effective battlers than Ozzie in Chrono Trigger.
Similarly, in both of the battles with the Shadow Sirens in Paper Mario: The Thousand-Year Door, the leader Beldam is the weakest member of the team compared to the far more deadly Marilyn and, in the second battle, the fairly more dangerous Doopliss.
Beatrix does all the heavy lifting in Final Fantasy IX, compared to Queen Brahne's incoherent rages.
Eirin Yagokoro of Imperishable Night does pretty much all the work while Princess Kaguya just sits around doing nothing, at least as far as fanon is concerned. Many fans also believe Eirin to be a more difficult boss than even the true final boss of the game.
Reisen gets shades of this plus a little Only Sane Man on the side...when she's not busy being the Butt Monkey of the Eientei group.
Similarly, Sanae Kochiya from Mountain of Faith is often portrayed as a much more mature character than the goddesses she technically serves.
UFO also brings us Nazrin, who is increasingly portrayed as much more serious and diligent than her "master" Shou, thanks to Shou canonically losing her Jeweled Pagoda, forcing Nazrin to find it for her.
Even earlier is the maid Yumeko, right-hand to Shinki. Sure, Shinki created the whole of Makai, but Yumeko single-handedly takes care of the day-to-day affairs of the whole of Makai. She is often compared against her Windows-era Expy Izayoi Sakuya, a human maid who serves the vampire Remilia Scarlet (though not an instance of this Trope).
Lin from Advance Wars: Days of Ruin is this to both Brenner and Will, being the tactical brains behind the operation while her superiors act more as The Heart. Lin is fine with working under them, and admires Brenner immensely, but she's also the single most competent and practical character in the main cast; coming up with the army's tactics, showing much more practical, if cynical, solutions to the problems the groups face than the others do, killing General Ripper Greyfield, and generally helping to stop the MASSIVE Hero Ball Brenner and later Will carry from causing too much trouble.
In Persona 3, Jin serves as The Dragon and hypercompetent sidekick to suicidal Nietzsche Wannabe/Dark Messiah Takaya, often restraining the latter when he decides on a whim to point a gun to his head to make a message or take on a large group of Persona-users quite capable of beating him with just four of its members.
Tron Bonne from Mega Man Legends. While her brother Teisel is seen as the head of their "Evil Family", she's clearly the one that keeps things moving. No wonder she's the protagonist of her own game.
In the Castlevania series, Dracula is often a pansy Anticlimax Boss, where as Death is more often than not one of the hardest bosses in the game, generally being much harder to hit, faster and having much better attacks. In 'Castlevania: Rondo of Blood, Maria is a little girl Ricther rescues, and playing as her makes the game MUCH easier. It's taken to an extreme in Castlevania: Lament of Innocence. Dracula aka Matthias relies on Death to do all the heavy lifting in the endgame such as stealing Walter's vampire soul and Death'' is the Final Boss.
Eddie Riggs in Brutal Legend, as the ultimate roadie serves this role to the Resistance, whose leaders lack The Power of Rock and Eddie's skill with stage building and handling of equipment.
Romance of the Three Kingdoms X is known for having a weak AI, even for high-stat leaders. As a result, if you're playing as a vassal instead of a ruler, it's entirely possible to have more troops in your district than all of your force's other districts combined.
Entirely possible in Castlevania Crusader Kings 2, if a character has low stats a council stocked with the best advisers in the realm is very helpful.
Genis from Tales of Symphonia is this to Lloyd. Tales of the Abyss has Guy. Granted that it's not hard to look hyper-competent compared to the 7-year old Luke, but Jade suspects that Guy is not a mere servant because he is too smart for a commoner.
Alexei Stukov is this to Gerard DuGalle in the StarCraft series. Du Galle has successfully managed to screw up nearly everything the UED has done, and is easily tricked by Duran. He even got tricked intohaving Stukov killed! Meanwhile, Stukov is very hesitant to follow his boss's orders, knowing what the outcome would be, and easily sees through Duran's deceptions. It's very obvious that if he were the one in charge, the UED would already own the universe. It's even stated in the manual that he's the smartest of the two. Kerrigan mocks DuGalle this way. She claims that Stukov was twice the man DuGalle was, and she's ever so grateful that DuGalle saved her the trouble of killing Stukov. The player's Non-Entity General fills this function throughout the campaigns, commanding forces for most of the plot important events in the story. This only applies inside the first game, in the books and the second game, these characters are assumed not to exist (except [possibly] the magistrate, who may just be hiding).
Jyunichi is this to Shogo Akuji in Saints Row 2. Even thinking in meta, Jyunichi's boss fight is a long katana duel, whilst Shogo's boss fight you just gun down some mooks and briefly chase him on a motorcycle.
From Sonic the Hedgehog, Miles Prower A.K.A Tails is often joked about as being this. He is a mechanical genius who built a fake (but almost perfect) Chaos Emerald (with one key difference) in Sonic Adventure 2 and as a matter of gameplay in Sonic Adventure had to out pace Sonic by using his flight capabilities, indicating he can. Added to that is his swimming is much better than Sonic's and his occasional invulnerability.
In Sonic the Hedgehog 2 and 3. Not only is he every bit as fast as the protagonist, he can also fly (taking Sonic with him in the 3rd game), swim, and is completely invulnerable to enemy attacks, to the point of not even losing rings when struck. In addition, any rings or powerups he collects automatically go to Sonic, regardless of the distance or obstacles between them. Parodied in Awesome the Hedgehog with this delightful exchange:
Robotnik: Now, Sonic will fall right into my trap!
An interesting variation in Skies of Arcadia occurs with Aika, the Lancer to Vyse. The game has a Swashbuckler Rating, which gives a title to Vyse and affects his reputation, ranging from the ability to recruit crew members to prices at shops (the higher the rating, the lower your prices). There are But Thou Must trees in the game that can increase, decrease, or not affect the rating. Making a crapload of bad choices sends your rating down (the lowest you can go is Vyse the Ninny), and whatever Vyse says in those makes Aika this trope. Whatever Vyse suggests in a good choice (i.e, "shall we sneak into the Coliseum" or "save everyone right now") will be suggested by Aika in a bad choice.
Otacon has been called more effective than Snake at everything that Snake does, despite his tendency to urinate in fear.
Stern of Magical Girl Lyrical Nanoha A's Portable: The Gears of Destiny, what with being the Material of Wisdom to the Evil OverlordRoyal Brat Lord Dearche. She's the one who inevitably calls most of the shots amongst the Materials since her logic easily trumps Lord Dearche's childish orders, a fact that annoys her lord a lot.
In Breath of Fire III and IV, she really is a Physical God. In IV she's stuck inside a suit of armor and the main character is a free god, so it's a bit less jarring.
Prince of Persia (2008): Elika is the Double Jump, the source of your magic attacks, can do all the same acrobatic moves you can [without his trumped up glove], and is the one who heals the land. Oh, and she saves him from certain death all the time. The only thing the Prince has that she doesn't is a sword. Justified in a sense in that she planned to save the land by herself. The prince is more of a Supporting Protagonist.
A radio show on Fallout 3 featured "adventurer Herbert Dashwood and his stalwart ghoul manservant, Argyle." In each episode Herbert's bumbling landed them in situations Argyle would singlehandedly extract them from. The player could meet Herbert in game, who claimed he wasn't really that big of a fool but admitted Argyle was most certainly a badass.
You can find Argyl's corpse later. He fell trying to defend a village from slavers in a battle Dashwood barely escaped from, the news of which will crush Dashwood.
Fallout: New Vegas: Just about every companion can be this if you play them right. Give them the proper equipment and do their quests to obtain the right perks, and they will crush armies with you. The game also gives one to the player in the form of Boone. In particular for Speech/Science/Barter style characters, the best way to win in combat is to hide behind cover and wait for Boone to kill everything.
The module The Light Reborn for Neverwinter Nights has you literally play the sidekick to the actual hero, he tackling the main enemies while you go on the smaller errands. This all culminates into a giant "tipping the scale" moment that's even based on a player's earlier decision and can thus fail. Never has being the sidekick felt so rewarding.
In The World Ends with You, Joshua is none other that Shibuya's Composer. Even though his powers are diminished, he manages to kill some Taboo Noise, which are the hardest to beat in the game, with a single blow.
There's a failed, to an extent, attempt to avert this in Ancient Domains of Mystery. A necromancer or a bard can easily have an extremely powerful sidekick (or a few!), in extreme situations rarely ever having to fight at all. The penultimate level, however, is a Slippy-Slidey Ice World, so you can't take any sidekicks with you. But... if you take a Scroll of Familiar Summoning with you, you can summon a new friend in the final level. And the power of a familiar depends on the so called "danger level" of a dungeon. Cue a Greater Earth Elemental of an insanely high level mopping the floor with the several boss monsters guarding your objective.
The Flash game Help The Hero is all about this. While the hero is the one that fights monsters and gets all the glory, the player is his sidekick who has to make sure that he's properly equipped for said fights or else end up beaten to a bloody pulp.
The main character of Radiant Historia is one of these in one of the two timelines. While his commander and best friend Rosch is fairly competent on his own, Stocke is The Ace, and thanks to Time Travel often comes across to other people as Crazy-Prepared and nearly prescient. Stocke is just fine with this, since he feels Rosch is a much better leader, and when he's nominated for a command position turns it down with a rather vehement no.
In Fire Emblem: New Mystery of the Emblem ~Heroes of Light and Darkness~ a unit called the "My Unit" is created at the beginning of the game, and is the eponymous "Hero of Shadow." The thing is, no other character in the Fire Emblem series has been so powerful. This character is given a unique class selection with amazing bases (the Myrmidion My Unit starts with TWELVE speed at level ONE. Compare Athena, the Myrmidion you get at the Prologue, who has the same amount of speed at level TEN), custom growths, the ability to reclass and alter these growths to become even better (unlike Marth) and an eight-part prologue that just fills him/her with ungodly amounts of experience, and, since the chapters in the prologue can be completed in a minute or so, a strategy for the higher difficulties is to reset the game until the My Unit gets the growths the player wants. The reason he/she counts as a Hypercompetent Sidekick? They are the "Hero of Shadow," meaning, in the history books, Marth gets all the credit for the My Unit's achievements. This becomes incredibly jarring when in the later chapters, the My Unit is running (or flying, as reclassing into a flying unit is entirely possible) around, killing everything while Marth is running around visiting the villages and trying not to die.
While Kondou in Hakuōki is not particularly incompetent, it's his second-in-command Hijikata who does most of the work of leading, planning strategy for, and maintaining discipline within the Shinsengumi. The negative consequences of this become clear after the Battle of Toba-Fushimi; Hijikata's efforts to protect Kondou from the harsher realities of leadership leave him unprepared to cope with the Boshin War and unable to retain the respect of his men when things start to go wrong.
In the first Mercenaries game, there is Josef, a Russian mafia liutenant flown from Moscow to North Korea with the express intention of supporting Sergei, the exceedingly incompetent local mafia boss. With the help of the main character, Josef exceles at reluctantly making his boss' insane plans come to fruition. Being so hypercometente actually works against him, as Sergei grows increasingly paranoid about Josef planning to usurp his position (which wasn't even the case, as Josef honestly felt honor-bound to obey his boss no matter what).