It is a curious thing, Harry, but perhaps those who are best suited for power are those who have never sought it. Those who, like you, have leadership thrust upon them, and take up the mantle because they must, and find to their own surprise that they wear it well.
Yang Wen-Li of Legend of Galactic Heroes would rather be a historian, but he is really, really good at the whole "admiral" thing. The Alliance government does not understand this, and grows increasingly distrustful of him out of fear he will make a power grab. This eventually results in his death.
One Piece: King Dalton of Drum Island used to be the right-hand of the deposed tyrant, King Wapol, after his wiser father died. He's only the ruler of Drum because the citizens keep voting for him to stay as king, despite hoping that they will one day learn to depend on themselves.
Gan Fall also counts. After being reinstated as "God" of Skypiea, he laments that he won't be able to enjoy planting his pumpkins anymore.
In Archie Comics' Sonic the Hedgehog, both Sally and Elias are pretty much of the same mind, but for different reasons - Sally wouldn't mind the throne, but she'd rather have it on her own terms: no mystical mumbo-jumbo and preferably with Sonic by her side. Elias plays the trope straight, not wanting to be a ruler and only took up the throne to save the kingdom from Patch, Antoine's Evil Counterpart. When told that his stepdaughter would be ineligible, he was quite fine with that.
Hawke in the Dragon Age IIfanficThe Whole Sky Shaken is pushed by his friends and people around him to run for the Viscount, since he’s one of a few people who can stand up to Meredith and make a difference. He doesn’t really want any position of power and prefers to kill things with fire rather than get involved in the politics. When he finally does decide to go for the throne Anders blows up the Chantry and they both have to flee Kirkwall.
Maximus from Gladiator, although he, of course, doesn't get to rule.
In the Lord of the Rings movies, Aragorn refuses to pursue the throne until he absolutely has to to save Middle-Earth. In the original books, taking the throne was on his to do list... but not very high up.
In The Name Of The King: A Dungeon Siege Tale has Jason Statham's character Farmer turn out to be King Konreid's long-lost son Camden, whom everyone thought dead. The current heir to the throne of Ehb is the king's nephew Duke Fallow, who isn't fit to rule, to put it mildly. However, after being told the truth, neither Conreid nor Farmer are willing to believe it. Farmer is bitter about the king's army not being there to protect his wife and son from the Krug, and his only concern is freeing his wife and friends from Gallian's clutches, the kingdom be damned. Konreid finally persuades him to reconsider after imparting on him the importance of the kingdom having a good ruler (i.e. not Fallow) and Farmer realizing that he needs the army to help him defeat Gallian.
The Icarus Agenda by Robert Ludlum has a conspiracy of Well Intentioned Extremists who want to give America a good President, implying one who didn't seek the power for its own sake, but recognise that one such would be unlikely to volunteer for high political office or to beat the more power-hungry and politically wily candidates if he did; their solution is to find a suitable candidate and manipulate his career without his knowledge.
Random from The Chronicles of Amber is probably the only one of Oberon's children who never seriously contended for the throne of Amber, and it was him who was given it by the Unicorn in the end of the first cycle. As the second cycle shows, the trust was well-placed.
A downplayed example in The Traitor's Hand. Cain remarks that the local planetary regent, being the chief bureaucrat rather than the usual aristocratic appointee, is considerably more sensible than the average governor. He suggests that "the absolute last person who should end up with power is the one who wants it".
Both Emperor Rene and King Alexander in the Arcia Chronicles hate their jobs but are placed among the best sovereigns Arcia ever had. Subverted with Anhel the Luminous, though, who actively pursued the throne, started a rebellion, and is remembered as a great ruler. Then, subverted again, in that the historical perception of the just rulers is twisted by the Church, whereas Anhel actually did enough evil in his life to be reborn as the mightiest among the Horsemen of the Apocalypse.
In Reflections of Eterna cycle by the same author, the only guy who is reasonably suitable to take the Taligian throne in the face of the impending cataclysms, Roque Alva, has so far managed to ingeniously evade this obligation for six books in a a row. Yeah, it seems like a recurring motif of the author.
Danny Saunders in The Chosen who does not want to be a Rebbe.
Emperor Gregor from the Vorkosigan Saga attempts to bunk off from being emperor during one of the early books. He's mostly scared by the history of insanity in his family though.
"Welcome, Prince," said Aslan. "Do you feel yourself sufficient to take up the Kingship of Narnia?" "I — I don't think I do, Sir," said Caspian. "I'm only a kid." "Good," said Aslan. "If you had felt yourself sufficient, it would have been a proof that you were not."
In Julian May's Galactic Milieu series, the job of Dirigent (ruler of a planet) is traditionally given to someone who doesn't want it.
In Megan Whalen Turner's The Queen's Thief series, Eugenides, the Thief of Eddis, marries the queen of Attolia because he's in love with her, but he doesn't want to be king.
"He didn't marry you to become king. He became king because he wanted to marry you."
In the John DeChancie science fiction comedy book, Living with Aliens, it is revealed that all of galactic civilization runs on this premise. The Federation figured out that people who want power should never have it. Their solution is to find the people who want to be rulers the least and force them, against their wills, to rule over the galaxies. This leads to them running away constantly and spending all their time getting high and abducting people from earth to molest and probe them for fun.
In The Tamuli, the Isle of Tega builds its entire government on this. Nobody asks to be nominated for public office; as soon as you're nominated, you're placed under guard, and if elected your possessions are sold and the proceeds put into the treasury. At the end of your five-year term, if the economy prospered, you get your money back with proportionate interest. If it floundered, you could lose everything. Many officials have worked themselves to death for the good of the republic.
In The Republic, where Plato writes that the philosophers who have no interest in ruling should be forced to become kings for the benefit of the people. The gist of it is "We want someone who has to be dragged into office kicking and screaming, but who will then do the best job he can in hopes of being let off early for good behavior."
Jenna Heap in Septimus Heap is reluctant to become a Queen at first, but she grows more accepting over the books.
King Cinhil Haldane in The Legends of Camber trilogy. He had been a cloistered priest for many years when he was found and removed from his monastery. He was persuaded to give up his vocation and accept a wife and a crown, but he didn't like it, and he came to blame Camber for his misery. Centuries later, Prince Regent Nigel Haldane is this in prospect; he prepares for his empowering ritual with a heavy heart because he doesn't want to confront the possibility he'll become king, and he insists on postponing his coronation for a year when his nephew is reported missing.
King Zachary Hillander of the Green Rider series. He was the younger son and expected to govern Hillander Provice while his elder brother Amillton ruled, but King Amigast finally saw Amillton for what he was.
Lord Vetinari of Sir Terry's Discworld does not rule the city because he wants to but rather because he feels it is his patriotic duty.
Considering A) the sort of people who preceded him and B) The Chessmaster could be renamed "The Vetinari" with little to no change of meaning, he might have something there.
Carrot is probably the heir to the throne of Anhk-Morporknote the books have never provided unequivocable proof, but the circumstantial evidence is pretty strong, and enough people think he is that whether it's actually true is neither here nor there. but refuses to claim the position of king. He would be a great king but feels that it would be for the wrong reasons (he wants people to do the right thing because it is right and not because a king ordered it). He'd probably accept the position if there was no other way to save the city, but it hasn't been necessary as yet.
Nobby is actually offered the position of king of Anhk-Morpork but runs away from the offer not because he would not want the job but because he is afraid of what Commander Vimes would do to him. He would make a horrible king.
Come to think of it, Sam Vimes wasn't exactly wild about becoming Duke of Ankh, but that's more to do with his dislike of hereditary privilege; wielding quite considerable authority in his role as Commander of the City Watch is a different matter.
Nick Seafort in the Seafort Saga, who desperately tries to get out of being promoted to Captain, desperately tries to get out of executing people, desperately tries to get himself killed despite his crew's attempts to save him, and commands anyway because he believes it's his duty to the UN and to his unmerciful God. Basically subjects his crew to the wrath he thinks he deserves.
Raamo in the Green-Sky Trilogy is actually terrified by the prospect of becoming one of the ruling Ol-Zhaan, to the point of trying to telepathically scream at the people looking at him with awe that he's just an ordinary Kindar. It's his reluctance that tips off Neric and sets the bigger plot in motion. later, we learn that D'ol Falla had intended this as a Batman Gambit. The only one who could set things right was someone who wouldn't be caught up in the trappings of power.
The Man In The Shack in The Hitchhiker's Guide to the Galaxy subverts this to a deliberately extreme extent, in that he's not so much reluctant as unaware that he even is a ruler, and apparently is unwilling to accept that there's anything to rule in any case. In fact, the people who gave him that power decided that anyone who actually wanted to be in charge couldn't be trusted with the job, so the guy who makes all the decisions needs to be someone who doesn't even know the rest of the universe exists.
The Roman Emperor Claudius, at least the way he's portrayed in I, Claudius and Claudius the God.
Titus Groan, in Mervyn Peake's Gormenghast novels (as well as the 2000 BBC miniseries), is anything but honored to be the 77th in the long line of the Earls of Gormenghast and will do everything he can to try and get free of the duties of his rank.
In Sisterhood of Dune, the newly-created Sisterhood (precursor to the Bene Gesserit) on Rossak realizes that an heir of Emperor Salvador Corrino will become a disastrous tyrant and ruin the Imperium. They decide to secretly make Salvador sterile, so that his more competent brother Roderick would continue the imperial line. This nearly works. Unfortunately, the Sister administering the drug lets it slip to Roderick that he would make a great ruler. Unfortunately for them, Roderick has no desire to be Emperor and is fiercely loyal to his brother. He has the drug examined and the Sister arrested and interrogated. The brothers then arrive to Rossak and execute a number of Sisters, including the remaining Sorceresses.
In A Song of Ice and Fire, Robert Baratheon didn't rebel against the Targaryens because he wanted the Iron Throne. He did it because he wanted to take back Lyanna Stark from Rhaegar Targaryen and because Rhaegar's father Aerys had ordered his and his best friend's Eddard Stark's deaths. At the end of the rebellion, Robert killed Rhaegar but felt no satisfaction since Lyanna died under mysterious circumstances anyway. Robert was forced to take the Iron Throne due to the Baratheon line being descended from a Targaryen bastard. Robert was an incompetent ruler because he hated being stuck with an uncomfortable throne made of melted down swords, a wife he hated because she wasn't Lyanna, and the huge responsibilities of ruling an entire continent which he passed on to his advisors.
And then he appointed Eddard Stark to be his Hand, thus making him the one who had to de facto run the Seven Kingdoms. Ned's bad feeling he had upon taking up this position was eventually proven right: His honorable attitude was not all too well suited for handling the Deadly Decadent Court of King's Landing - in fact, it did eventually result in Ned's death by execution.
This is a major part of Perrin's Character Arc in The Wheel of Time, the other part being his coming to grips with being able to talk to wolves. He spends most of the series resisting leadership, and when he finally accepts it reflects that perhaps those who least want authority are most deserving of it.
Live Action TV
Angel: About the only thing Groo fails at is running a country; He became bored with the committees, and was deposed pretty quick.
Star Trek: Deep Space Nine: When it becomes clear just how corrupted Gowron's become, to the extent where he's willing to sacrifice the war against the Dominion just to get rid of political rivals, Martok is forced against his will to replace Gowron as Chancellor strictly speaking he replaces Worf as Chancellor.
Babylon 5: ex-Narn Ambassador G'kar is both stunned and hesitant when told he has become a religious icon to the Narn, and outright refused leadership of the government after overthrowing the Centauri. He eventually accepted spiritual, if not political, leadership, but only because the public wouldn't take no for an answer.
Claudius in I, Claudius becomes this after he's proclaimed Emperor by the Praetorian Guard following his nephew Caligula's assassination. He really wants to make Rome a true republic again, but the Guard literally forces him into the imperial role and Claudius realizes he has to accept or risk having himself and his family slaughtered. Much later, however, he decides to marry his corrupt niece Agrippina and let her unhinged son Nero succeed as emperor, to force the Romans to get sick of monarchy. He only succeeds in insuring that his only son and only loyal adviser die.
In Once Upon a Time, King Leopold proposed to Regina because she was the only woman he'd met since his wife's passing who genuinely cared about Snow. Young Regina never wanted to be Queen and only wanted to marry her true love, her family’s stable boy. Inverted in that Regina’s mother killing her true love and forcing her to marry the King is what made her become, well, the Evil Queen.
In LOST, character Jack Shepard is a doctor and tries to save lives after the airplane crashes into the island. The other characters begin seeking his advice on all matters and treating him as their leader, though no formal decision is made that he is in charge. Despite the fact that Jack does not want this responsibility, the other characters look to him to be their leader and save them from crises. He struggles with the burden but ends up being the stoic leader even though he has no desire to do so because nobody else is suitable for the job.
In Noob, Arthéon succeeded to Master Zen after he stopped playing not as much because he wanted to be Guild Master as because the two other members of the guild were a Man Child and a self-centered guy who is convinced that letting female players into the guild can only weaken it.
Fate/stay night: Saber hates having been made king, having never wanted it but doing it anyway for the good of her people. She did the best anyone could, but her dislike of being king at all (and the ensuing lack of drive) meant she was never well-loved by her followers or her people and the Dirty Business she was forced to do as a king eventually led to her kingdom tearing itself apart. She wants the Grail so she can wish for a better king to take her place in history. In Fate/Zero, Rider delivers what is effectively a "The Reason You Suck" Speech to her about it, claiming that her lack of desire for kingship meant she was never truly a king, just a "young girl playing at being a king".
Final Fantasy VI: Neither Edgar nor Sabin particularly wanted to rule Figaro, so they settled it with a coin toss, where the winning twin got to do whatever he wanted. It's eventually revealed that Edgar used a double-headed coin so that Sabin could get the freedom he wanted.
Alistair of Dragon Age: Origins can become this. He can become considerably less reluctant about it if you select certain dialogue choices however, he proves to be a much better king after said dialogue choices than otherwise.
Sebastian Vael in Dragon Age II is another take. He envied the fact that his older brother would inherit the throne and spent most of his early life as The Hedonist. His parents put him in the Chantry to stop him embarrassing them, but he found it gave him a purpose. Then political rivals murder the rest of his family, and he spends the game trying to choose between retaking his lands or serving the Maker. You can push him one way or the other.
"When I wanted to rule, I would have been terrible at it. Now that I might be decent, I don't know if it's the right thing to do."
A non-government example: Squall from Final Fantasy VIII hates leading but is forced to lead an entire army when they're attacked, and does a nice job while at it. Rousing Speech and everything.
According to the Saints Row instruction manual, Julius claimed to be this when he started leading the 3rd Street Saints, though the Stillwater Police believe he's a natural gang leader.
The Protagonist in the sequel, Saints Row 2 however, is a subverted example, being happy with leading the Saints in Julius' place instead of just 'keeping his/her mouth shut and letting himself/herself being told what to do.' He/she even outright states this when he confronts Julius, who was revealed to have rigged the boat to explode in the first game.
Peony in Tales of the Abyss. He's the illegitimate son of the previous emperor, and when all his half brothers get killed he's called to the capital. But, because he is going to be the Emperor, he can't marry the woman he loves. All his subjects adore him.
Subverted with Martin Septim in The Elder Scrolls IV: Oblivion: had he actually become The Emperor, he would very likely be even better than his father, since he never wanted to become one in the first place. Additionally, his experience as a dark priest and later, a poor monk would certainly make him care for his subjects a lot.Alas...
In Heroes of Might and Magic IV we have Lord Lysander of Palaendra. After Enroth was destroyed by the Reckoning, strange portals opened to a new world called Axeoth. The survivors of the kingdom of Erathia found themselves without a nation and a ruler, as every Gryphonheart (the ruling dynasty) is believed to have been killed by the Reckoning. Seeing that no one else is willing, Lysander forms the new nation of Palaendra. He becomes such a great ruler that people offer to crown him king. He refuses, however, as, in his mind, only a true Gryphonheart can be king. He always considers himself merely a steward. However, when a man shows up claiming to be a Gryphonheart, Lysander is reluctant to hand over the throne. Not because he isn't willing, but because he doesn't trust the newcomer's claim. When Lysander ends up proving that the man is an impostor and finding out that he himself is a Gryphonheart, he reluctantly allows himself to be crowned king.
In Knights of the Old Republic II, if Mical (aka the Disciple) was not corrupted by a dark side Exile, Kreia reveals that he will help rebuild the near-extinguished Jedi Order and "sit upon the new Council, reluctantly, as all good men do".
In the Grand Finale of Avatar: The Last Airbender, Zuko wants Iroh to take his rightful place on the throne and is very hesitant to face his destiny as the next Fire Lord, but accepts his responsibility and is crowned at the very end. This is in contrast to his father, who usurped the throne and eventually crowned himself king of the world. He continues to feel this way throughout the comic sequels, to the point that in Avatar: The Last Airbender - The Search his initial reaction when he finds out he might not really be Ozai's son and thus not a legitimate heir to the throne is relief.
The Sword in the Stone: Considering how Arthur became King or how he lived until then, all reluctance can be excused.
In the Codename: Kids Next Door episode "Operation I.T.", it's revealed that nobody wants Numbuh 362's job as Supreme Leader of the organization, and when she starts hating the job herself because of the stress and decides to quit, the whole organization holds a game of tag, which whoever is "it" at the end of the time period having to take the job. (Unfortunately, when Father ends up in control of it via a technicality, Numbuh 362 has to show true leadership skills to get her job back, which include eating through broccoli to fight him, a sacrifice that puts her in a coma for two weeks after she manages to defeat him. And she still takes back the job afterwards.)
Ramiro II of Aragon was the fourth son of King Sancho Ramirez, and consigned to the church. Unfortunately, his elder brothers died young and without heirs. He was dragged out of the church under protest, married a widow who had an established record of healthy births, stayed around just long enough to sire a daughter, then betrothed the daughter to a trusted ally, and marched back to the church.
George Washington, the first President of the United States of America, did not want the position in the first place, and outright refused to be made king of the new country when the idea was presented.
He also refused to serve more than two terms, starting a tradition for the office. It was made law by the 22nd amendment, ratified after Franklin D. Roosevelt was elected to four terms.
William Howard Taft was another very reluctant President, having been pressured into taking the position by his friend Theodore Roosevelt and to a lesser extent, his wife. He was a judge at heart, and had little taste in politics. (His ideal job would have been a Justice of the Supreme Court, which he eventually achieved.) Hardly any other President was so unhappy in the White House, and worst of all, his arguments with Roosevelt during the 1916 campaign destroyed their friendship. At Wilson's inauguration, he told the new President, "I'm glad to be going. This is the loneliest place on Earth."
In another Chinese example, Liu Bei of The Three Kingdom's fame, did not like the idea of titling himself as King of Shu Han as he was loyal to the (now defunct) Han Dynasty, whom he aimed to restore. His advisors, particularly the famed Zhuge Liang, advised him by taking the title of King of Shu Han, he is able to oppose the other two kingdoms better in his attempt to restore the Han dynasty
Furthermore, Zhuge Liang reminded him that he is a Liu, the family that built the Han Dynasty to begin with, which means he had a more legitimate claim to call his Kingdom of Shu as Shu Han
Prince Albert, Duke of York, would have liked nothing more than to live quietly with his nice family, and, since he was the second son, this was a reasonable expectation. However, his brother Edward VIII turned out to be a complete screwup, a playboy and Nazi sympathiser who earned the contempt of the establishment and was forced to abdicate. Albert (as George VI) shouldered the duties of the King-Emperor, the stress of which is often thought to have considerably hastened his death, and he's generally considered to have done so pretty well (being the figurehead for the nation during World War II didn't hurt).
Technically, he may not have counted as a "ruler" (depends on your definition), but Pope Benedict XVI may not have ever wanted to be Pope, claiming publically that he prayed he would never gain the position. (Unfortunately for him, in the Roman Catholic Church, one oath all priests much swear to upon becoming ordained is to be willing to accept the position if selected.)
Yao, the first king/emperor of China, supposedly passed his throne to Shun because he was the most competent and he didn't want the job. Shun similarly passed it to Yu, but Yu couldn't find anybody who qualified who wasn't too old, and so when he died, the throne went to his son and the position became hereditary. Future emperors would try to justify their own rule by having the previous regime (if said ruler wasn't related by blood) "voluntarily" abdicate.