One 2013 Toyota commercial shows a family finding a genie and getting various wishes, and the daughter wishes she was a princess. Cut to the princess in armor on a tiny horse in front of an army, raising her sword and telling the army to "Avenge my father's death!" and charging the castle. Her then very-much-alive father cheers on from the sidelines with "Yeah, avenge my death!"
Olivier Mira Armstrong, the Four-Star Badass in charge of the Briggs Fortress, is this without actually being royalty; in the manga, her men are shown to occasionally refer to her as "the Princess" or "our Queen".
Lyrical Nanoha has Olivie Segbrecht, a.k.a., Sankt Regina Olivie, a.k.a., the last Sankt Kaiser of the Cradle. In life, she was known as the queen deemed second to none in combat. In death, she was known as the Sankt Kaiser who gave her life to end the Ancient Belka War, and worshipped as a Christ-like figure.
Nefertari Vivi, princess of Alabasta, from One Piece, who leaves her country and infiltrates a group of bounty hunters, making her way up to the top brass, to discover the identity of the ringleader who's planning a coup against her father; lost her cover, she joins the Straw Hat pirates to fight openly against the usurper, and finally manages to stop a civil war by SCREAMING "STOP" IN THE MIDDLE OF A BATTLEFIELD. She's not at the level of the main heroes for strength, but surely she doesn't lack guts.
Theodora from the Hellas Empire doesn't actively fight, but she does try directing the action too.
Maria Louise from G Gundam evolves from Tomboy Princess into this when she volunteers to destroy a cornerpost rodeated by a lethal forcefield via a molotov cocktail.
Although NOT action-oriented, To Love-Ru's Princess Lala of Deviluke is no slouch in combat with her main weakness being her tail.
Ayeka and Sasami of Tenchi Muyo! fit here. While Ayeka isn't as powerful as Ryoko and Tenchi, standing up to Kagato when the latter was thought dead isn't something to laugh about. Then again, Nine times out of ten, she's arguing with Ryoko, so… As for Sasami, in the OVA, discounting her bond with Tsunami, she's shown to have a proficiency for staffs great enough to disarm a high-ranking Galaxy Police officer and enough skill to evade said officer's traps. In Tenchi in Tokyo, she was given control of Ryo-Ohki this time around, which dumps its OVA-only humanoid transformation for a mecha transformation, which Sasami shows to really work on the initial Monster of the Week that showed up.
Relena Darlian/Peacecraft of Mobile Suit Gundam Wing combines this with Badass Pacifist. Stands up to the biggest Badass of the series? Check. Stands up to her own brother? Check. Stands up to the group that wanted to manipulate her and in turn manipulates them? Check.
Sailor Moon, as well as the rest of the Sailor Senshi/Sailor Scouts, who are eventually revealed to be princesses in their own right.
Princess Sapphire, the eponymous Princess Knight, is likely the Trope Maker in manga and anime. Having been made back in The Fifties by Osamu Tezuka, the series is commonly cited as the main influence for many female writers who would eventually revolutionize the shoujo genre and make this a popular character archetype.
Punie from Dai Mahou Touge who is unstoppable with her magic… and a lot more dangerous if you manage to use an Anti-Magic weapon on her. Her battle cry of 'submission is the princess way' doesn't mean she'll submit to you. It means she knows every submission hold that can break your bones.
A Certain Magical Index has the second royal daughter, Carissa, who possesses an impressive understanding of military strategy and tactics.
Magi - Labyrinth of Magic has a few - Princesses Hakuei and Kougyoku of the Kou Empire, the former of which being a general in the army. Kougyoku starts off as a villain and both are King Candidates. Princess Dunya Mustashim, of the Mustashim kingdom, is another villainous example.
Seikoku No Dragonar has the Lautreamount sisters of the Lautreamount Knight Country: Silvia the fourth princess who is a Dragonar and Veronica who once killed 100 men without getting injured. Eco, Ash Blake's dragon who takes a form of a young girl, is the Imperial Princess of Avalon; she once fended off a group of assassins who were sent to capture her, defeating most of them.
Li Chang Ge (a.k.a Princess Yongning of the Tang Dynasty) from Choukakou is definitely this, with a Guile Hero flavor.
Princess Adrienne from Princeless. Not only does she decide that doesn't need a prince to rescue her, but she actively mocks a number of the racist and misogynistic tropes present in most fairy tales.
Princess Kavatah in Megalex. She leads her troops into battle personally.
Iolande of the Green Lantern Corps was this, being a princess who was chosen to be a Green Lantern. Unfortunately, she had to take up the mantle of Queen since she was the sole survivor of her world's royal family. Her advisers refuse to let her go In Harm's Way, limiting her chances to be badass.
An interesting case in Dungeon Keeper Ami - after she becomes Empress Ami might qualify, as a small, slight girl of fourteen. She has received combat training, however, as a prominent element of several story arcs. Her adopted sister, Tiger, manages to fall even further outside the bounds of this trope, as she is essentially a being of pure magic and anthromorphic...well, tiger. Not to mention prominently muscled and inclined to a crass, or at least Genki Girl, personality.
The My Little Pony: Friendship Is Magic fanfic Rites Of Ascension has Princess Luna as this. Her sister claims that Luna is probably the single most dangerous thing on the planet short of Discord, and that’s while she’s still in the process of recovering from getting hit with the Elements of Harmony.
Mulan is easily the poster girl of Badass Disney Princesses, with this being a key basis of the premise itself. She's a slight subversion because she's not a princess, but she's considered part of the princess lineup nonetheless.
Ariel from The Little Mermaid was probably the first badass Disney princess and was most certainly the first to save her prince. The TV series, sequel and her appearance in the Kingdom Hearts video game only solidified her badassery. Ariel was the only female party member in the first game (unless you count Tinkerbell, who was a summon), and knew powerful Fire/Ice/Lightning magic, among others.
In the Pixar film, Brave, Mérida is more than willing to take on a bear that's over twice her size. She eventually decides that she won't let random guys she's never met compete for her hand in marriage. She also has enough skill with a bow and arrow to put Katniss Everdeen or Hawkeye to shame.
In The Lion King, Sarabi leads the lionesses who do the hunting. However, it's only mentioned when there's no food to hunt. There's also Nala, who almost kills Pumbaa and always pins Simba when they wrestle.
Princess Fiona from the Shrek movies shows definite badass tendencies, such as fighting off bandits with kung fu and headbutting Prince Charming. In Shrek Forever After, Alternate Universe Fiona gets tired of waiting to be rescued, so she breaks out and becomes leader of La Résistance. The other princesses in Shrek the ThirdTook a Level in Badass to fight Prince Charming. Especially Snow White, who uses her power to summon woodland creatures to assault Charming's minions. Also, the Queen shows Fiona from which side of the family she got her fighting skills by headbutting through two walls. She doesn't feel so good after the second wall, though. Understandable since she's getting on in years.
Princess Mononoke doesn't control an actual kingdom, she does have her own family of wolves and fights pretty especially with a spear.
Rapunzel from Tangled also counts, even if she doesn't know she's a princess for most of the movie.
Vanellope von Schweetz from Wreck-It Ralph casually confronts a giant Punch Clock Villain most people are scared of, repeatedly outsmarts The Caligula and his guards who (fail to) hunt her down like an animal, and eventually becomes a Badass Driver with the help of the previously mentioned Punch Clock Villain. She is eventually revealed to be a princess, and she turns down the title to become a president instead. Vanellope is all badass.
Princess Anna in Frozen smacks down a wolf with a lute, fends off a giant Snowlem, sacrifices her life for her sister, and punches her ex-fiance in the face, sending him flying over a railing. She is most assuredly badass. Her sister, Queen Elsa, has ice powers that can kill people via impaling or giant snowmen. Luckily for everyone, she doesn't actually go through with killing people.
Dragons Fire And Ice. What does female lead Princess Kyra do when she's surrounded by evil orcs? Simple. She kicks their asses. Or burns them.
Maria Posada from The Book Of Life is a Badass who's also the daughter of General Posada, the leader of San Angel.
Princess Leia started as the Badass in Distress. She even managed to take out a stormtrooper before getting captured. Through most of the series, she is a prominent and useful fighter; even Go-Go Enslavement doesn't stop her from using the chains binding her as weapons.
She gets it from her mom, former elected Princess of Theed and Queen of Naboo. Padme was in the thick of it during not only the battle of Theed 10 years before the Clone Wars, but during several engagements during that conflict, including Geonosis. She fought in and survived a battle that killed off dozens of Jedi.
Éowyn from The Lord of the Rings is the niece of King Théoden, the granddaughter of his father King Thengel, and rides into battle taking out the leader of the Ringwraiths (a task Gandalf failed to accomplish), although he was destined not to fall to the hand of any man. Emphasis on the man part. Though not technically a princess, she was close enough to count in a rough sense, especially as Rohirrim were a "heroic" society; they may not have been as legalistic about aristocratic terms as one which had evolved into a Vestigial Empire. Also after the death of Théodred, her brother is named Théoden's heir. And at the end of the trilogy she marries Faramir, and Aragorn names them Prince and Princess of Ithilien. She's decided to be a Retired Badass at that point, but it still counts.
Lúthien from The Silmarillion is specifically a princess, and while she doesn't physically fight, between her magical abilities and pure grit (staring down the Physical Gods of Evil and Death in succession just cements it) she more than qualifies as Badass. The Quest of the Silmaril - assigned to her beloved Beren as the requirement for being able to marry her, and involving stealing one of the most valuable items in Middle-earth from the most powerful being in Middle-earth - is accomplished pretty much entirely by Lúthien, with Beren contributing comparatively little.
Mathilda Arminger of the Emberverse rides with Rudi Mackenzie's war band, and is one of the most effective swords at his back.
Jessica, due to her Bene Gesserit upbringing; Fremen leader Stilgar even worries when he first meets her that she'd end up ruling his people (the toughest warriors in the known galaxy) through Asskicking Equals Authority alone. She's technically not a princess, but she's the natural daughter of a Baron, concubine of a Duke, and mother of one God Emperor and grandmother of another one.
Her daughter-in-law Chani is technically this to the Fremen, as the daughter of Liet-Kynes and every bit as tough as any full-blooded Fremen man or woman.
Her granddaughter Ghanima would be a literal example, especially after surviving an assassination attempt involving remote-controlled tigers.
Princess Thirrin Freer Strong-in-the-Arm Lindenshield, Wildcat of the North from The Icemark Chronicles definitely counts. Not only can she hurl a throwing ax with the best of the Housecarls, but she battled an army of hundreds of thousands led by the wily General Scipio Bellorum and won! She manages to persuade murderous vampires, werewolves, sentient trees and bigass cats to help her in the war, without being eaten!This isn't even mentioning her other moments of badassery from the sequels, either...
Bronwyn from Bronwyn's Bane. She's expert with a sword.
The title character in the innovative feminist Fairy Tale classic The Paper Bag Princess, who rescues her prince from a dragon. He then refuses to marry her because she doesn't look beautiful anymore—she had to improvise an outfit, and she tells the ingrate to buzz off and goes off to live her own life.
Snow White in The Sisters Grimm series teaches a self-defense course whose members are called "The Bad Apples".
Susan, Lucy, and Aravis in The Chronicles of Narnia, though the first two are Queens and the last one is minor Calormene nobility (who later becomes Queen of Archenland).
Parodied in one of the stories in the Chicks in Chainmail anthology, where training academies have sprung up specifically because Badass Princesses are in style. Tough luck for the very, very Princess Classic protagonist.
Princess Thayet jian Wilima in Lioness Rampant. She's skilled with a crossbow and, along with her bodyguard, takes it on herself to guard a group of refugee children to the safety of a convent.
While most of her badassery was done before Ozma made her a princess, Princess Dorothy is still considered the Land of Oz's heroine and champion. She's still pretty strong-willed and feisty even with a crown on her head. Princess Ozma herself is a delicate and dainty thing, but very much a badass sorceress when circumstances call for it. Neither princess takes very kindly to anyone threatening the peaceful citizens of Oz, or each other.
In Fiona Patton's Tales of the Branion Realm series, gender is no object, and royal women are knights and do a lot of fighting. In the first book, the Crown Prince's betrothed brings her weapons and squire on their tour of the country and helps fight off an attack on the party. Later, she becomes very angry at him over her pregnancy, because it means she won't be allowed to go with him to war.
The Firebird Trilogy: Firebird, youngest daughter of the Netaian queen, had a choice between going into advanced music studies or going into the military. She chose the military, and has repeatedly demonstrated her fighting skills and courage. She also has no diplomatic skills, contrasting with her sisters, Princess Classic Carradee and Politically-Active Princess Phoena.
King Arthur's daughter Rhianna in Katherine Roberts' Pendragon Legacy series.
The Xanth series, where pretty much every female character is a badass in her own right.
In Fool, the re-telling of King Lear from the view of Pocket, Lear's "Black Fool," it's revealed that this story's version of Cordelia ends up as one, conquering many lands, including Spain. Why is Pocket stunned by this when he learns it? Because "she was rubbish at chess." Real war, he thinks, must be easier for her.
Sister Fidelma is a fictional sister to a historical King of seventh-century Ireland. As a religieuse, she doesn't go armed but is still capable of defending herself in hand-to-hand with a now-obscure Irish martial art.
The first is Abigail Hearns, Miss Owens, daughter of Steadholder Owens. Abigail is a petite woman with waist-length brown hair, who is consistently described as "cute as a button". Raised on Grayson, a backwater planet where women are traditionally not expected to do anything other than be wives and mothers, Abigail browbeat her father into letting her study every hard science course she could get her hands on. When she finally came of age, Abigail was the first native-born Grayson woman to go through Saganami Island, Manticore's Naval Academynote Grayson has its own, Isaiah Mackenzie Naval Academy, but at the time Abigail went to Manticore it was in the middle of having its curriculum entirely revamped and was also far too small to handle the explosively growing Grayson officer corps. She passed Saganami with flying colours and graduated sixth in a class of eleven thousand. She is currently serving as the Chief Tactical Officer on a brand-spanking-new modern destroyer, on the fast-track for independent command, and quickly making a name for herself as one of the best tactical minds of her generation.
In the back-story, Emperor Gustav Anderman VII definitely qualifies. When her older brother, heir to the throne of the Andermani Empire, was killed, the male line was extingusished and it seemed the Empire would descend into petty dynastic squabbles between her cousins. Instead of letting that happen, she marched her bodyguards into the Chambers of Parliament, forced Parliament to legally declare her a man and eligible to inherit the throne at gunpoint, and essentially dared her cousins to try to make their complaints stick. Gustav VII is to this day the longest-reigning and best regarded Andermani Emperor in history.
All the female members of the Valdemar Royal Family in Heralds of Valdemar. Also Princess Idra of Rethwellen who renounced her place in the succession to become a mercenary captain.
A Mage's Power: Princess Kasile of Ataidar takes lessons in both martial arts and combat magic. She's not as good as professional soldiers, but she can take care of herself fine.
In the Nightrunner series, the nation of Skala has a divinely-mandated matriarchal militarist monarchy. Badass Princesses are therefore the norm, and early in the series the reigning Queen's middle daughter is considered ineligible for the throne, unlike her older and younger sisters, because she has chosen to eschew military pursuits and be a socialite. The eldest sister is a general and the youngest a cavalry commander. This leads to a fair bit of intrigue, because nobody is stupid enough to think that killing or capturing them would be easy.
Shannara has had a few, most of them from the Elessedil family:
Elf Queen Of Shannara reveals that Action Girl Wren Ohmsford is actually Wren Elessedil, granddaughter of the current Queen of the Elves, and her designated successor. Having grown up with the Rovers, Wren can more than take of herself, and her discovery that the Elfstones in her possession are real allows her to take on even the Shadowen with ease.
High Druid Of Shannara introduces Khyber Elessedil, younger daughter of the Elven King, apprentice Druid, and current wielder of the Elfstones. She's one of the few Druids to stay loyal to Grianne Ohmsford after Shadea a'Ru's coup, and joins the quest to free her. By Dark Legacy of Shannara, she has taken Grianne's place as High Druid, and become a genuinely frightening Lady of War, whose advanced age barely slows her down. It takes Tael Riverine to bring her down, and even he has to cheat.
Aphenglow Elessedil, granddaughter of King Emprowen Elessedil was introduced in Dark Legacy, where she was forced to learn how to fight when Edjina Orle came for her and her sister. By The Defenders Of Shannara, she has succeeded Khyber as High Druid and Lady of War.
Live Action TV
Xena: Warrior Princess is something of an odd case, considering Xena was called a princess, but wasn't of noble birth and never ruled a kingdom. Gabrielle, on the other hand, is an actual princess of the Amazons.
Lady Morgana from Merlin fits this trope, especially after the season three reveal.
Delenn in Babylon 5, being a member of the Grey Council for part of the series, generally not afraid of anybody or their armada, and being a descendent of Valen. And being a Badass.
Daenerys Targaryen is a princess of the Targaryen family, who used to rule the Seven Kingdoms. However, the family was overthrown, so she and her brother are not technically royal in that aspect. She marries a Khal (warlord) of the Dothraki and becomes his Khaleesi, wherein she turns into a powerful badass and leader. She hatches some dragon eggs and becomes commonly known as "Mother of Dragons," a reference to both her dragon-hatching and her royal ties to the Seven Kingdoms and Targaryen family. She attempts to take the Seven Kingdoms back from the current rulers.
Also Arya Stark, the daughter of a lord who becomes regent after King Robert dies. She is a very good archer and swordsman, and is depicted throughout the series as a tomboyish rebel. Eventually, she runs away and disguises herself as a boy, going around kicking ass.
Myth and Religion
Medea of Classical Mythology. She qualified as princess by being the daughter of a king, and qualified as badass by cutting her brother into pieces when he tried to take her away from her fiancee Jason. That was without her magic. With her magic she killed an unkillable bronze giant who threw rocks at her ship. You really wonder why Jason thought he could dump her and get away with that...
In a later Yu-Gi-Oh!Duel Terminal storyline, Emilia is known as the Dance Princess of the Nekroz. Her special effect prevents the negation of Nekroz ritual spells and protects Nekroz Ritual monsters against card effects that target them. She also can turn into the Nekroz of Gungnir.
Sonic the Hedgehog: Blaze the Cat, essentially a pyrokineticLady of War from another dimension with some seriousissues. As if that wasn't enough, her version of Eggman is said to be much crueler than Sonic's, and unlike Sonic, she has no friends to turn to for assistance, at least not until she meets Sonic and his friends.
Princess Natalia from Tales of the Abyss is very proficient with a bow and arrow, accompanies Luke and his pals on their journey, helps solve political issues, never gets kidnapped, (okay, maybe once, for about ten seconds) and stands for her nation no matter what.
In the fourth game, there's Princess Ayra, with her punishing "Shooting Star Sword" ability; she even threatened to kill her "master" if he ever double-crossed her and her charge Shannan. Also, Ayra's daughter Larcei, who has all of her mother's skills and might pick up good ones from her father as well (depending on who the player chooses for that role), as well as Princess Tailto of Freege, though she meets a really sad end.
Princess Raquesis and her daughter Nanna. They both start as a weird mix of Action Girl and White Magician Girl, but can properly become this with training.
Princess Briggid of Jungby is a subversion, as she was kidnapped as a child and became a Pirate Girl. She never 100% takes up the princess role; after reuniting with her sister Princess Adean... well, shit happened, which left her amnesiac and traumatised for years, though she eventually regains her memories.
And then we also have Princess Altena, a Lady of WarDragon Rider from Thracia. Actually from Manster, as she's the long-lost sister of Prince Leaf of Lester.
The Archanea games feature Caeda/Shiida, the original badass princess from the series. She is the princess of Talys and serves as an invaluable addition to Marth's army, both as a pegasus knight and as a Guile Hero who can recruit many allies to Marth's side. The games also feature Princess Minerva, starting a trend in the series of a female dracoknight who joins late in the game.
Lyn from Blazing Sword, who is both the princess of the Lorca tribe in the Sacae plains and the heir to the throne of Caelin in Lycia.
Once she throws away the Distress Ball, Princess Lilina from Binding Blade becomes a Magical Girl Warrior version. There's also Sue of the Kutolah tribe in Sacae (who can also potentially be the daughter of the aforementioned Lyn).
In Awakening we have Lucina, potentially the most powerful character in the game, plus her possible sisters Cynthia or Kjelle, neither of whom is a stranger to combat. Lissa also can become this after being promoted or reclassed, while Say'ri of Chon'sin is badass right from the get-go. And if we count princesses by marriage, then all of Chrom's potential wives will count, since none of them will stay at the Ylisstol palace when it's time to go to war with Valm.
Morgan also qualifies if she is the daughter of Say'ri or Lucina.
The title character of the series spent its first entries getting rescued, but has become increasingly badass as the franchise has gone on. At the end of Ocarina of Time, Princess Zelda is revealed to have been disguised as a freakin' ninja, then helps unlock barriers during the escape from Ganon's Tower, and even pins him down while Link delivers the fatal blow. In The Wind Waker and Twilight Princess she battles alongside Link with her light arrows for the Final Battle, and in Spirit Tracks, Zelda possesses Phantoms, actually fights beside Link, and is even playable for certain segments.
Midna, the titular Twilight Princess, definitely qualifies as well, given that she helps Link out during combat, paralyzing a group of enemies in an energy field, leaving them weak and defenseless while Wolf Link attacks them. And then there's the times she used the power of the Fused Shadows to trash Zant, or threw Beast Ganon around.
Yuffie from Final Fantasy VII is the daughter of the Lord of Wutai, and also happens to be a ninja. Who wields a giant shuriken as a weapon, among other large, throwing-oriented pointy things.
Although she acts like a Princess Classic throughout Final Fantasy IX, it is Garnet's idea to escape from her castle, trek across thousands of miles of dangerous territory, and seek political assistance in stopping Queen Brahne.
Alena of Dragon Quest IV can't get her father to approve of her adventuring, so she kicks down her bedroom wall and leaves that way. Something she's clearly done before, given you have to Talk to Everyone in the castle to give the repairman enough time to board it up before making your escape. She even has to save her own fake double at one point of her chapter.
Super Princess Peach is this trope. Mario and Luigi are captured by Bowser, and Princess Peach has to save them. This is a complete inversion of the typical storyline of many Mario platformers. Princess Peach in general has been getting a lot of this lately, especially in the RPGs. Let's count:
Rosella of Daventry from King's Quest: Willing to become a Human Sacrifice to save her homeland from a dragon in her first appearance, inspiring her long-lost brother to fight said dragon. In the next game, she takes on a quest in a dangerous and hostile land to save her dad. This involves dodging cave trolls, ogres, a triad of witches, a poisonous snake, escaping from a whale, and then killing a wicked fairy. Her second game? Rescuing an imprisoned king, attacking a were-bear, digging her way out of a volcano, and snapping her boyfriend out of Brainwashed and Crazy.
Lady Cousland and Lady Aeducan, the Human Female Noble and Dwarven Female Noble respectively, who are two of the potential player characters in Dragon Age: Origins. Lady Aeducan is the daughter of the dwarven King Endrin, and thus rightfully a Princess of Orzammar. Lady Cousland's father is a Teyrn, which is more on par with a Duke, but a codex in the game notes that her home region of Highever is actually a principality. The Couslands don't use princely titles, but even without them, they are second only to the actual royal family in terms of the peerage. Also, Lady Cousland is the only player character who can potentially end the game by becoming Queen of Ferelden.
Though not exactly a princess, the Krogan female known as "Eve" in Mass Effect 3 is the only volunteer who survived a series of gruesome experiments to restore her species ability to reproduce, wears a veil, is very soft spoken, and needs to be escorted to an escort ship. When taken from her pod to step into the shuttle, another group of attackers come around the corner, to which she just grabs Wrex's shotgun and blasts them all away, stating that she doesn't need babysitting. She's still a Krogan after all.
Liara, being the daughter of a highly revered leadership figure withing Asari society would count then, too. After all, she could flay you with her mind.
Arguably Tali as well. She's the daughter of Migrant Fleet Admiral, highly respected political/military figures that run Quarian society. Although she starts off as more of a Wrench Wench than an Action Girl, she Took a Level in Badass between 1 and 2.
Yggdra Union: Although she becomes a Distressed Damsel during an episode, requiring her friend Milanor to go rescue her, Yggdra Yuril Artwaltz fits the trope during all remaining chapters, leading the Fantasinian Royal Army in the retaking of her kingdom, then the invasion of Bronquia, from the frontlines. In between these two campaigns, she even becomes a Badass Queen.
Sveta of Morgal, who among other things hides her status from the heroes for fear they'll think she's involved in the sticky political situations of the plot. Once trouble starts, she joins the party to help end the Grave Eclipse and atone for it on her brother's behalf, culminating in an attempted Heroic Sacrifice… and she's one of the most powerful player characters in the game, even without counting her beast form.
Reco from the Mushihime Sama series goes through four Bullet Hell games without appearing to break a sweat. In Bug Panic, when Kiniro is unable to help her, she just picks up miniature versions of her bombs and deals with the enemies on foot.
From Perfect Cherry Blossom is Yuyuko Saigyouji, a nobleman's daughter turned ghostly princess of the Netherworld, and one of the most broken characters in the setting due to her ability to kill with a thought.
Kaguya Houraisan of Imperishable Night is a princess from the Moon, exiled to Earth for drinking the elixir of immortality - the same princess from the Tale of the Bamboo Cutter, in fact. Her powers over eternity allow her to shred the protagonists' spell of endless night after she's defeated.
Afterward, Kaguya sends the heroines to battle her equally-powerful immortal rival, the fire-flinging EX-boss Fujiwara no Mokou, who also qualifies as the daughter of Fujiwara no Fuhito, one of the most powerful noblemen of late 7th/early 8th century Japan.
Silent Sinner in Blue gives us the Watatsuki sisters. Toyohime has what's implied to be a Wave Motion Gun in the form of a fan, and feels confident in a confrontation with Yukari (and is apparently a Reality Warper in her own right, though unlike Yukari she can only manipulate reality on the moon). Yorihime can summon gods, and managed to take out several powerful characters in succession. Fandom hates them both. Especially Yorihime.
Ten Desires' Toyosatomimi no Miko is a weird case, since she's the same person as the legendary Prince Shotoku. Though female, she's referred to as Crown Prince by her followers Mononobe no Futo and Soga no Tojiko, who as high-ranking ladies from early Japan may also qualify as Badass Princesses.
Double Dealing Character introduced Shinmyoumaru Sukuna, a kobito princess who helped start the game's Incident in hopes of creating a world where the weak wouldn't get picked on by the strong. Due to her size she's easily the most Badass Adorable of the lot.
Kairi from Kingdom Hearts seems to be on her way to becoming this. She's established as one of the seven Princesses of Heart in the first game. Towards the end of Kingdom Hearts II, she's shown fighting Heartless with a Keyblade given to her by her childhood friend Riku. Birth by Sleep reveals that she got her Keyblade wielding ability from an encounter with Action Girl Aqua as a child. And presently, in the secret ending of Dream Drop Distance, Yen Sid sent for her specifically to train her in combat in order to defend herself against an upcoming threat.
Melia Antiqua from Xenoblade, the crown princess of the High Entia. In gameplay she's the party's Squishy Wizard, having the least HP out of all the characters but being capable of inflicting some serious damage in the right circumstances; while in the story she shows several instances of badassery, like severely hurting the beast that has just killed her guards single-handedly, completing a trial that had killed many of her ancestors, and turning the tides of seemingly lost battles quite a few times.
In the Dark Parables series, Princess Briar Rose (Sleeping Beauty) becomes this by the fourth game in the series, going from a very understandable Damsel in Distress in the original game to a strong and confident mage who wields powerful plant magic.
Princess Sophia of the Awakening series of PC games. What makes her especially badass is that in her world, every human has some ability to use magic. Magic attacks affect only those with magic and since Sophia was born without this ability and relies solely on her brains and wits, she's immune to such attacks.
Battle Princess Of Arcadias actually has MULTIPLE Badass Princesses, who serve as the Kingdom's strongest fighter. Interestingly, at least one part of the sidequest levels imply that they aren't Princesses by heritage; rather, they're made Princesses upon clearing a test(which involves lots of fighting). Main Character Plume is the Battle Princess of Schwert Kingdom, fellow playable character Violon was a candidate in the same test(s) that Plume passed to become Battle Princess, and they mention a Battle Princess for Armatura as well. Albeit the Armatura Princess is said to have been slain by Luis when she turned traitor...
Wild Arms XF has Alexia Lynn Elesius who is an expert at swordplay, she wields a BFS, and she's pretty much a Mighty Glacier in-game.
Princess Ariella, the warrior princess of Dwarf Country, from Nefarious.
Princess Waltz, as the name implies, is loaded with princesses who are total badasses.
Higgs: Where, I'm guessin', you were used to bein' one o' the toughest things around...
Now that her pupil Agatha is the ruler of Mechanicsburg, Zeetha is eager to instruct her in the ways of being a princess.
Zeetha: First lesson, every princess needs a Battle Axe. Here, use this one until we can find something more impressive.
Agatha: Ah. That kind of princess.
Though it hasn't been openly stated in the comic yet, all the clues suggest that "Chump" is the Baron Wulfenbach himself, making Zeetha princess of more than just Skifander.
Tower of God: All of Zahard's Princesses, bar Repellista. They are adopted because they are naturally powerful (or as with Repellista, gifted), they get a Power-Up by receiving Zahard's blood and then proceed to take a level in badass. Or rather, 134 levels. Except Repellista.
Princess Pi constantly wins battles and overcomes enemies with no assistance from others. It helps that she was born physically invulnerable.
In Heart Core, we have our protagonist Ame Lashiec, the Princess of Asgard who is a powerful magician and was capable of defeating most of the Overfiends singlehandedly.
Aisling the Erl-king's Daughter in Roommates is very sweet and looks and acts like a classic princess. Just don't forget that she is a powerful fae or make her mad (she was able to intimidate a magical mercenary with a freakinghair pin for threatening her favorite ship).
Silver Serpent of the Whateley Universe. Her father is the dreaded Asian supervillain Iron Dragon, and she is apparently the first child of his to have mutant superpowers.
Also, as of winter term, Jobe Wilkins, now the superpowered female child of the King of Karedonia. When stuck in a holding cell full of angry female supervillains, Jobe beat the entire cell into submission.
RWBY's Weiss Schnee is an "heiress, actually" but still fits this tropenote being a crown-wearing Winter Royal Lady whose family is very wealthy and politically powerful. In her trailer she takes down a hulking armored knight with nothing but her Semblance and an Aura-channeling fencing blade. In the series itself, Weiss's martial skill is said to be matched only by her poor attitude... and she has a very poor attitude.
In Aladdin: The Series, Jasmine elevates to this standard to the point where she actually participates in a few battles. Most notably, she opens a can of whoop-ass on Mozenrath after he kidnaps Aladdin. Even within the movies, King of Thieves has Jasmine fighting against said thieves fairly well more than once.
Princess Calla, in Disney's Adventures of the Gummi Bears, fits the trope. Her father holds a tournament of arms to determine which of his knights will be his daughter's protector. She secretly enters the contest - and wins. Furthermore, her father finds out what she did and is deeply impressed enough to publicly proclaim she needs no protector for herself.
Zahara from Spider Stories is both a warrior and a whiz kid. Her stance says it all.
While Toph is not exactly a princess, she still is the only child of a very prominent noble family in the Earth Kingdom. Toph is a fantastic Earthbender who is the only one who can bend metal, is almost completely unhampered by her blindness, and fights in underground Earthbending tournaments to pass the time. There's a reason why it's said "Toph isn't blind, she just can't see anything less awesome than herself"!
In X-Men: Evolution, Storm turns out to be the princess of an African tribe. Does that demote her to a helpless Damsel in Distress? No! She's still the same Storm we know and love. Given that the comic book version was worshiped as a goddess by her tribe, that's no surprise. This is just a backstory that's been modified a little so as to avoid offending Moral Guardian types who would find it blasphemous that a human character was treated as divine.
Teen Titans: Starfire is actually the princess of the planet of Tamaran. Long after she leaves her royal family behind, she remains strong and vigilant. And since Blackfire is her sister, she would be another evil example.
Princess Ilana in Sym-Bionic Titan. Not only does she have a personal suit of Powered Armor and co-pilot the title mecha, she's the first one to run off to fight a gigantic monster sent to kill her to. A monster made of living fire that's been obliterating the city singlehanded. She also demonstrates in the season 1 finale that even if she's not on the same level as Lance, she's still no slouch in hand-to-hand combat.
From the same creator: Princess Gwenevere, heiress to the throne of New Avalon and her Archenemy, Princess Kale [her aunt]. Kale took up dark sorcery to Take Over the World. Gwen and crew are all that can stand up to Kale's attempts, especially after Kale took out Merlin.
The main character of She Ra Princess Of Power, aka Princess Adora of Eternia. While she did have the option of returning to her home world to rejoin her royal family, she chose to remain on Etheria, fight monsters and every now and then spin-kick Hordak through walls.
As demonstrated during the season 2 finale, threaten her subjects, and she wont hesitate to try and fry your ass with a Frickin' Laser Beam. Even if she did lose, definite badass points, and the only reason that she lost was because the villain had received a massive power boost.
Princess Cadance handles herself pretty well, too. Even when not using The Power of Love to send Queen Chrysalis flying like a kite, she managed to hold up a magic barrier around the entire Crystal Empire. Even exhausted to dangerous levels, she didn't falter. And when she gets recharged with The Crystal Heart? Well...ask what's left of King Sombra.
And now we have Princess Twilight Sparkle, who's already known to be quite badass, and certainly doesn't become less so after her coronation.
If "Twilight's Kingdom" is any indication, the combined power of the four princesses is roughly equivalent to that of every pony in Equestria combined plus Discord's(!) And Twilight, with the help of her friends, unlocks and uses a power which is even stronger that.
An episode of G1 of My Little Pony features six princess ponies, who destroy the lava monster Lavan at the end, thus making them six of the seven characters to actually kill a villain in G1, the other being Megan.
Princess Lotta Lamour from the TaleSpin episode "The Road to Macadamia". She protects her kingdom from an Evil Chancellor and his Mooks by attacking them with a big mallet.
Princess Zelda from the 1989 cartoon The Legend of Zelda. She did most of the fighting and had to save Link many times.
Bloom, Stella, and Aisha/Layla in Winx Club are the princesses of Domino, Solaria, and Andros respectively. They are also powerful fairies that frequently fight and defeat evil witches, wizards, and monsters. In the 4Kids version, Musa is the princess of Harmonia, and she's part of the same team as Bloom, Stella, and Layla
In Gawayn, Princess Gwendolyn is surprisingly kick-ass when she wants to be, despite being shrunk to about six inches tall.
Lakshmibai, the Rani of Jhansi in India, who fought for the independence of her realm against the British in India and was killed in battle during the Mutiny.
Aethelflaed, The Lady of the Mercians: the daughter of Alfred The Great who joined with her brother to inflict a defeat upon the Danes. Example: The Danes attacked her wedding party. She not only organized an effective defense, she rallied the guests and led a counter-attack which drove the attackers off.
Princess Anne of England foiled her own kidnapping attempt by shouting, "Not bloody likely!" to her would-be abductor, but only considered hitting him. She managed to escape unscathed, while a bystander was the one to slug the would-be kidnapper and save her. They even made a TV movie about it.
Kahena, the leader of an alliance of non-Moslem berbers against the Arab invasion.
Gwenllian, a 13th-century Welsh princess who lost her life defending Kidwelly Castle from a Norman invasion.
While married to Louis VII, King of France, Eleanor of Aquitaine insisted on taking part in the Crusades as the feudal leader of the soldiers from her duchy. After that marriage fell apart she married Henry II, King of England. She tried to join their three sons in the revolt against her husband but was stopped by the King on the way and held in captivity.
Princess Pingyang is an example, although she wasn't technically a princess until after she raised an army to put her father on the throne.
Kurban Djan Datkha, a Kirghiz chieftainess who led Caucausus tribesmen against the Russian invasion in the nineteenth century.
Most XVIII-century Russian empresses were former Badass Princesses who succeeded in grabbing the crown by force. There also was one who didn't succeed.
Katherine of Aragon, Queen of England and first wife of Henry VIII. Though often portrayed as "the chubby one who couldn't have a baby", she more than deserves the title of Badass Princess. She was named Queen Regent and ran the country while he was away - and did so very capably. While Henry was off fighting the French, for example, she managed to utterly defeat a hundred thousand Scots as they invaded England. They must have thought that the country would be a pushover, what with the king and most of his army being overseas. Cut to Katherine sending Henry the torn, bloodied coat of the King of Scots. Ouch.
Back when she was styled Princess of Spain and Dowager Princess of Wales, she was also the first European woman to serve as official ambassador to a foreign court. Her father, King Ferdinand, got rid of the former ambassador at her insistence, and when he didn't have an immediate replacement, he appointed his daughter to do the job. Despite being Henry VII's daughter-in-law, Katherine had been treated very poorly between the death of Prince Arthur and her marriage to Henry VIII (the death of Queen Elizabeth of York left her without any sort of protector in the court), but she turned out to be a skilled politician. During the first decade and a half or so of his reign, Henry VIII would not do anything without her input.
Another queen of England, Margaret of Anjou, wife of Henry VI. Because Henry was an unfit ruler prone to fits of insanity, Margaret had to effectively rule the kingdom in his place and was the de facto leader of the Lancaster side of the Wars of the Roses. She personally organized the Lancastrian armies and was responsible for several victories before the Yorkist armies began to claim victory. Undaunted, she continued to try to rally troops and fought for the rights of her husband and son until the Battle of Tewkesbury in 1471, when her son was killed, which broke her spirit and caused her to give up her cause. Unfortunately these actions caused her to gain a reputation for ruthlessness.
Princess Olga of Kiev, who with a combination of cunning, ruthlessness, and plain taking advantage of people underestimating her for being a woman, managed to devastate and conquer the neighboring principality of Dereva who had killed her husband. As Cracked put it when they mentioned how she was canonized as a Saint despite her brutal achievement for helping spread Christianity:
Also an example of Princesses Rule, because medieval Rus was made of principalities rather than kingdoms.
Marie de Courtenay, daughter of the would-be Latin Emperor of Constantinople, Philippe de Courtenay, and his wife Yolande of Flanders. She was sent as a bride to her family's enemy, Theodoros Laskaris, emperor of Nikaia, and convinced her husband to make peace with the Latin Empire at a time when it was at its weakest. After being widowed, Marie went to live at her brother Robert's court in Constantinople. After Robert's death, Marie stepped in as 'Empress Regent' for their 11-year-old brother, Baldwin, and ruled ably until her untimely death at age 25.
Only one FEMALE member of the British royal family has ever served in the military (males do so quite commonly)... Princess Elizabeth, now known as Queen Elizabeth II. While being a truck driver in World War II with a women's auxiliary unit isn't as outright sword swinging as some on this list, the fact is the woman that most remember as the perfectly polite lady who many Britons adore as some sort of grandmotherly figure can probably out-mechanic 95% of the people in the country (since the aforementioned trucks were WWII vintage British ones). Combine that with the fact that she has talked down burglars that broke into her room in the middle of the night, refused to alter meetings with someone merely because they were the target of multiple assassination attempts, responded to riots with "calmness and courage in the face of the violence", and has apparently kept up with every report that crosses her desk (which are largely the same as the ones on the Prime Minister's desk) for 60+ years. Tony Blair once said he was more nervous about his weekly meetings with the Queen than he was about his weekly meetings with Cabinet.