In Animorphs, Hork-Bajir are a race of Noble Savages who are generally about as smart as a human toddler, except for "seers", genetic abnormalities with above-average human intelligence. Toby, the first free Hork-Bajir born in generations, is a seer, and none of her people question her leadership of the tribe. And note that Hork-Bajir age faster than humans, so even by the end of the series, she's just reaching adulthood at 2-3 Earth years.
The Beast Player: Halumiya ascended to the throne at only five after the deaths of her mother and grandmother.
After Emperor Justinian was blinded in a coup attempt a third of the way through The Belisarius Series, Empress Theodora avoided a succession crisis (by law, emperors must be whole in body) by adopting the eight-year-old son of her best friend (who, incidentally, also happened to be the stepson of the Roman Empire's best general) and putting him on the throne. Of course, setting herself up Empress Regent and naming her husband Grand Justicar were not to be sneezed at.
In Harry Harrison's Bill the Galactic Hero novel, the fleet admiral is an infant. His unintelligible utterings and cries are "translated" by his nannies into orders. No wonder the humans are having such a hard time beating foot-tall lizards. Then again, that entire universe runs on the Rule of Funny.
In Guilty Pleasures, the first book in Laurel K. Hamilton's Anita Blake Vampire Hunter series, a cruel and tyrannical child ruler, known as Nikolaos, serves as master vampire of the city, though she is technically one thousand years old. She reflects a villainous version of Mina Tepes, the ancient child-bodied master vampire from Dance in the Vampire Bund (mentioned in 'Anime').
The Chronicles of Narnia primarily have the Pevensie siblings who lead Narnia's Golden Age starting when they were 13 (Peter), 12 (Susan), 10 (Edmund), and 8 (Lucy). Likewise, Caspian is 13 when he takes the throne.
In Robert A. Heinlein's Citizen of the Galaxy, Thorby is the child ruler of an inter-Galactic mega-corporation, displacing a regent who is (secretly) unwilling to hand over power, then going incognito to get things done, etc.
Evading this is part of the background of the Codex Alera series, as Tavi's status as heir to the throne of First Lord is deliberately hidden for as long as possible to give him time to develop the skills he would need as ruler. By the time he learns his true name is Princeps Gaius Octavian, he's already had a full education at the most prestigious university in the nation, trained as a spy and assassin, and commanded a field army against impossible odds fighting an immense army of centuries old, two-meter tall wolfmen and won. By the time he's in a position to lead, he's managed to reach his mid-twenties.
Conqueror: Xuan at the end of Lords of the Bow is proclaimed Emperor of the Chin at the age of seven, after Zhi Zhong murders his father.
Cradle Series: Queen Sha Miara of the Ninecloud Court is one of eight Monarchs in the world, and is still a child. This leads to a lot of questions, as Monarchs are by far the most powerful people in the world, and it typically takes them centuries to earn that power. A brief view of the Ninecloud Court in the epilogue of Underlord makes it clear that Sha Miara isn't merely pretending to be a child, she really is one, and it takes her court a lot of effort to keep her whims in check.
At the end of Deltora Quest series one, Lief of Del takes the throne. He's sixteen, and thinks more with his heart than with his brain... and yet, because the books were inspired by JRPGs of the mid-1990s, this ends well. He grows over the story to understand that there is a time and place for emotions to rule. He thinks with both his heart and his clever mind. And, by the end of the series, he's 18 and has learned even more about evil, temptation, and duty.
In Katherine Kurtz's Deryni series, Kelson Haldane inherited the throne at the age of thirteen and was crowned the day after his fourteenth birthday. Also, King Javan Haldane, from the prequel series. Too bad he has an Evil Chancellor.
Several characters in Taylor Anderson's Destroyermen series, all female. Saan-Kakja, the Queen of the Fil-pin Islands, fairly quickly let her advisors know that she is the one who makes decisions on the isles despite her young age. Safir Maraan, the Orphan Queen of B'mbaado, also became a ruler of her city at a young age but remains not only a beloved queen but also a very capable warrior and general. Princess Rebecca Ann McDonald becomes the Governor-Empress of the Empire of New Britain in her early teens after her parents are murdered by a bomb planted by Lord-High Admiral McClain's people. One of her first acts as sovereign is to order the public execution of McClain, at the same time making radical changes to her nation's policies.
Downplayed in Jules Verne's Dick Sand, A Captain at Fifteen. Dick Sand, a fifteen-year-old sailor on a whaler ship, is forced to become captain when the rest of the crew dies trying to hunt one whale and he is the only one on the ship with sailing experience. Deconstructed when the ship's cook uses Dick's inexperience and that of the improvised crew to take the ship to Africa and enslave everyone.
The Boy Emperor of the Agatean Empire in Mort, who successfully poisons his Evil Chancellor and former tutor with the dish the chancellor prepared for him ("I had a good teacher."). Apparently later killed by his uncle, the Royally Screwed Up emperor in Interesting Times.
Teppic is still a teenager in Pyramids (just graduated from the Assassins-Guild) when he ascends to the throne of Djelibeybi.
Coin from Sourcery was ten years old when he asserted his power, becoming de facto ruler of the world for a time.
His father Leto Atreides became Duke at 14 after the death of his father Paulus in a bull ring. He rules justly and fairly. His first order of business? Banish his mother, who plotted to kill her husband.
Paul's children technically inherit the throne shortly after being born, but their aunt Alia rules as regent until Leto II seizes control when he's about 12.
The Empirium Trilogy: Though she's only twelve, Obritsa was declared the Queen of Kirvaya by the Magisterial Council. Justified in that the vote was rigged in favor of Obritsa, who had been raised by the human revolution to become the next queen.
The entire point of Ender's Game by Orson Scott Card is to train Andrew "Ender" Wiggin (who is a six-year-old boy at the start of the novel) to command humanity's forces in a Bug War. Apparently, Ender's older brother Peter and sister Valentine are just as smart as him, but they don't appear to have the qualities the fleet requires. Peter has all the makings of a serial killer (he psychologically tortures Ender and Valentine and dissects squirrels while they're alive), and Valentine has too much empathy for war. While Ender is at Battle School, Peter convinces Valentine to help him take over the world in order to prevent another world war, which is sure to start as soon as the buggers are defeated. They create online personalities called Locke (Peter) and Demosthenes (Valentine), which argue competing points with the eventual goal of having Locke having so much influence he can write policy for the entire world. By that point, the truth that they're really children will be known, but no one will care. And yes, this is exactly what happens. A short war breaks out after Ender beats the buggers (before he's 16), and Peter manages to stop it and is made Hegemon of Earth shortly after. Valentine's (or rather Demosthenes's) writings are still read and discussed thousands of years later. Oh, and Ender also manages to inadvertently start a new religion before he's 20.
It's not just Ender either; all of Battle School is made up of genius children being trained to be commanders of the fight against the Buggers. Several of the former Battle School students end up in positions of high power; Virlomi becomes the "Goddess" of India, Han Tzu becomes Emperor of China, Fly Mojo becomes leader of the Philippines, Alai becomes Caliph (though he's at least particularly a Puppet King), Bean Delphiki becomes Peter's right-hand man and helps him bring about the unity of the world, Petra Arkanian also works with Peter to bring about world unity and leads Armenian troops during the civil war on Earth and Suriyawong is given control of planning for the Thai Army and becomes Bean's ally and works along with Peter. The fervor they attract is so much that Peter convinces most of them to leave Earth just to avoid the inevitable conflicts they bring about, and even then it's with the promise of a colony, so they end up ruling in space as well.
In Graceling Realm, Bitterblue is about ten when she's crowned Queen of Monsea.
Greystone Valley has a 12-year-old girl serving as the land's main defense against an evil warlord.
In the Honor Harrington spinoff novel Crown of Slaves, 17-year-old Berry Zilwicki, posing as Princess Ruth Winton of Manticore, becomes the leader of a group of former slaves. By the time the deception is revealed, no one cares. The full version of the trope is barely averted, as her coronation as Queen Berry of Torch takes place shortly after her 18th birthday.
"I have a question for the grown-ups of America: Are you out of your minds? Are you expecting me to enforce the Constitution? I never even read it. I was absent from school that day. You want me as commander-in-chief of the armed forces? What if somebody attacked the United States? Would you really want me in charge? America must be in really bad shape if you elected me President. You better get it together and find some qualified people to run this country or we'll all be in big trouble."
In the sequel, The Kid Who Became President, he changes his mind and, as the title suggests, becomes the President of the United States. The deconstruction becomes even more poignant than the first, as he does a fairly terrible job until the end when he manages to avert a war using a video game. Afterwards, he again resigns, leaving his adult vice-president to become President in his stead.
Princess Ozma of the Land of Oz, who's roughly the same age as Dorothy (ten or thereabouts), is the benevolent dictator of what is essentially an empire made of four large countries and a city-state.
In the Liavek series, the Levar (hereditary ruler of the city-state) is ten years old in the first book, with all real power held by her regent.
Aragorn II, son of Arathorn, inherited the leadership of the Northern Dûnedain at the age of 2 when his father was killed by wolves in The Lord of the Rings by J. R. R. Tolkien. This is especially notable given that the Kings of Gondor and Arnor and their descendants (including Aragorn) normally live for well over a century and most do not even inherit the throne until they are over 100. Aragorn became King of Arnor (a somewhat empty title) at age 2 — which is the trope played completely straight — but he didn't become King of Gondor as well until he was around 90 ... which is perhaps not exactly a "child", but still relatively young by Gondorian standards. He proceeded to rule the (re)combined Kingdom for another 120 years after that.
At the end of The Obsidian Trilogy, we get the electoral version of this trope. After Arch-Mage Lycaelon Tavadon of Armethalieh resigned more or less in disgrace and what was left of the city's high council proved unable to agree on a successor from their ranks, a conclave of the all the mages in Armethalieh elected Cilarnen Volpiril Arch-Mage for both his valor in facing the demonic forces that nearly destroyed the city and his vast experience in dealing with the outlanders whose help was needed to restore Armethalieh's devastated lands. Cilarnen might have reached 19 at this point.
In Jack Vance's Planet of Adventure series: Traz. The nomad tribes' nominal leaders are generally teens or pre-teens—probably so the priests, who are the actual leaders, can have an easier time.
Happens quite often in A Practical Guide to Evil, when young Named ascend to positions that would otherwise be unattainable due to their age. Examples include:
Catherine, the Squire, who leads an army starting at the age of 16 and later becomes queen in all but name at age around 18,
Kairos Theodosian (the Tyrant) murders his predecessor and becomes king at age 13.
Michael Karl in Andre Norton's The Prince Commands ascends to the throne of a Ruritanian kingdom at the age of eighteen. When Ulrich Karl, his alleged dead cousin, shows up so that Michael Karl is not the heir, he's an even better example.
In The Psalms of Isaak, the Crimson Empress, ruler of The Empire of Y'Zir, is eventually revealed to be a toddler. Of course, though she's greatly revered by her people, she's not actually in charge of anything yet; Regent Eliz Xhum is the guy who actually runs the government. Oh, and the Empress' father, the immortal Wizard King Ahm Y'Zir, is still around too, but while he's revered as a living god by his followers, the fact that he's a senile, insane cyborg means he has little to do with actually running the empire he founded these days.
In Redouln Rising, Riley Bank and his fellow Redoulnite Leaders of the new generation rule over the kingdom of planet Redouln.
The Reynard Cycle: After the death of his father, Prince Lionel is named king at the ripe old age of three. He is murdered before he can turn ten. Next in line for the throne? A four-year-old-girl.
Gorath from The Riftwar Cycle, or more famously the RPG Betrayal at Krondor, becomes tribe chieftain at age twelve, and in subsequent years restores his tribe to strength and avenges their earlier near-massacre. Particularly notable because he's a dark elf, and would normally need to be at least a century old to be considered more than Just a Kid, let alone lead.
Sergey Lukyanenko's Seekers of the Sky: In Morning Nears, Marcus is a teenage bastard son of the ruler of the State with no claim to the throne. However, when it becomes clear that he may be the next messiah, everyone around him begins to think of him as their new ruler, which he will be once he gains enough supporters by making miracles. His father will be forced to abdicate.
Jenna is only ten years old when she is called to rule over the Castle in Septimus Heap. She doesn't officially become Queen until the last book, though, with former ExtraOrdinary Wizard (and ghost) Alther Mella serving as regent.
In Tanya Huff's Sing the Four Quarters, Lady Olinda sets up her nephew to be executed for treason so that she can control the new duc, his five-year-old son. The boy thwarts her in several ways before his father can reappear.
Naturally, more often than not Deconstructed in A Song of Ice and Fire. Being a rather realistic medieval Low Fantasy, children have to grow up a lot faster, both physically and psychologically, than in real life with disastrous consequences whenever they have to take positions of authority:
Joffrey Baratheon is a spoiled little sadist who is far more interested in using his power to feed his destructive urges than to do any actual ruling. Even before he became king, everyone saw him as a tyrant in the making and feared what he'd be like on the throne; the answer turns out to be "even worse than they ever imagined". He ordered Ned Stark's execution on a childish whim, which went completely against the warnings of his more politically-savvy family. This sparked a civil war that devastated much of Westeros. Eventually, he gets poisoned by his own allies at his own wedding at the age of 13.
Robert Arryn is a sickly, stunted boy who becomes Lord of the Vale at age 6 after his father's death. Robert is entirely dependent on his mother and not very likely to live long at all. His stepfather and bannermen are all vying for the title of Lord Protector so they can rule in his stead. The death of his mother at the hands of Petyr Baelish leads him to become emotionally unstable and he clings to Sansa as a maternal surrogate, who is 13 years old.
Jon Snow is made Commander of the Night's Watch at the age of 17. While he is just as honourable as his father, heroic, and quite a competent leader and strategist for his young age, very few of his older officers respect him. Compounded with mistrust from some due to his young age, his time with the Wildlings, his sympathy for the hardships faced by Wildling men, women, and children, and changes he makes in his efforts to save everyone from the oncoming army of the dead, he eventually gets betrayed and stabbed by a group of his own men and it's left ambiguous whether or not he survives.
Likewise, Daenerys Targaryen is a heroic character with the best of intentions and is another teenage ruler in this series, becoming the young Queen of Meereen. She struggles with rule when she becomes queen, tries to free the slaves, and utilizes some rather extreme methods for the sake of justice, leading to displeasing some around her. An assassination attempt is made on her life and after an incident in the fighting pits, she is believed by many Meereeneese to be dead, but in actuality, she escaped the city on her dragon, Drogon.
Robb Stark becomes the King in the North at age 15. Similar to his half-brother Jon Snow, he is honorable like their father, a young leader, competent in rulership for his age, and does his best to do the right thing but ends up displeasing some very important allies when he breaks a betrothal agreement. These allies — some of whom are his own men — betray and kill him in an event known as 'The Red Wedding'.
Tommen Baratheon is young, timid, and signs any decree thrown in front of him, making him quite a doormat and easy to manipulate.
A few of the Targaryen kings ascended to the Iron Throne as minors:
Jaehaerys I was crowned at 14, with his mother serving as regent for the two years before he reached majority; in contrast to many of the above young rulers, his reign was a long and prosperous one, and he himself was well-loved and respected by most.
Aegon III became king at only 11, with a council of regents ruling in his name for five years. Deeply traumatized by the events of the civil war that had led to him being the top claimant, he nevertheless survived to adulthood and worked to help the realm recover, though he himself rarely took joy in anything for the rest of his life. Due to his lost childhood, he was noted as seeming much older than his years to many of his subjects.
His son Daeron I was 14 when his father died, though he did not have a regency. A charismatic warrior prodigy, his priorities lay solely on conquering Dorne and completing the conquest of the continent that his ancestor had started, leaving most of the work of actually running the kingdom to his uncle. Before he hit 16 he had succeeded in this goal...for about a summer, after which it turned out that a talent for winning land on the battlefield did not translate to diplomatic skill at keeping its people from rebelling. Many thousands of lives were lost trying to reclaim Dorne before Daeron was assassinated only four years into his reign.
Daeron's brother Baelor I succeeded him at 17, though by Westerosi standards he was considered an adult.
At the time of Aegon's Conquest, the Kingdom of the Vale was ruled by a young boy named Ronnel Arryn, whose mother, Sharra, ruled as regent in his name. Sharra had to organize the Vale's army to fight the dragonriders, which was rendered moot when Visenya Targaryen managed to fly to the Eyrie and took advantage of a child's curiosity by inviting Ronnel to fly with her dragon Vhagar, an offer he happily accepted. Realizing that she had essentially taken him hostage, Sharra surrendered the Vale immediately.
All four books of Fiona Patton's Historical FantasyTales of the Branion Realm series feature young royals and sovereigns as protagonists, often leading armies. In the first book, the ruler is 24, and much younger for large parts of the book; his mother took the throne at 25 and led a "Charge of Carnage" at 14. In the second, the ruler is five; her two immediate predecessors both took the throne at age 10. In the third, the ruler is 18 and was twelve when he cunningly poisoned an entire enemy army.
Time Out Of Time: Timothy is officially named as the keeper of the market in book 2, "The Telling Stone". However, since he still has a life back home, he leaves Peter as interim keeper while he's away, while still popping in from time to time.
In Trickster's Choice, Dunevon Rittevon becomes king of the Copper Isles at about age four, with his tyrannical aunt and uncle acting as his regents. They eventually have him assassinated and take the throne themselves.
At the end, the rebels win and put Dovasary Balitang on the throne — she's thirteen. However, she intends to do the actual job, has no regents or puppet-masters, and adulthood is considered to start at sixteen in this setting.
Played with. Shushou, the Queen of Kyou, became queen at age 12 to save her ravished lands... 90 years before the story starts. And she still looks like a teenage girl, since she stopped aging after she ascended to the throne.
A bit straighter in Youko's case, as she's as much 17 and has just become the Queen of Kei.
When Luxa is introduced in The Underland Chronicles, she's eleven, and the Queen of Regalia. She became queen after her parents were killed by rats, but cannot assume full powers of the monarch until she's sixteen.
Young Miles counts for this in his role as Admiral Naismith, although most of his mercenaries spontaneously decide he is actually older than he appears because of an imaginary "Betan Rejuvenation Treatment."
In Warrior Cats: Sign of the Moon, Jayfeather realizes that Half Moon, who isn't yet considered a sharpclaw (an adult cat, to the Ancients), is the rightful cat to lead the Ancients and transform them into the Tribe of Rushing Water, due to her wisdom and her ability to read supernatural signs.The Sun Trail shows that Half Moon is still leader into her old age, and is considered great and wise.
Though not as young as most examples, at 18, Egwene al'Vere in The Wheel of Time is by far the youngest ever Amyrlin Seat. Her predecessor had been the youngest at fifty, and many who gained the Seat were several hundred years of age. The women who raised her meant her as a figurehead and referred to her as "the biddable child" in their planning, but she proved to be far more headstrong and savvy than any expected.
In The Windup Girl, 23rd century Thailand's current queen is a child - and that's about all we know about her, since she has a regent, or Somdet Chaopraya, ruling in her stead for the foreseeable future.