On 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!"
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.
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 Day 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.
Princess Diana of Themiscyra, a.k.a. Wonder Woman, who happens to be a member of the Justice League.
Teen Titans: Princess Koriand'r of Tamaran, a.k.a Starfire. Her wicked sister Komand'r (Blackfire) also counts.
In the X-Wing Series comics, Plourr Ilo, a butch-looking female Boisterous Bruiser, turns out to be the last surviving member of Eiattu V's royal family. She never stops being badass.
Princess Sally from the Sonic the Hedgehog Comics (and their accompanying ''Sonic Sat AM animated series) leads the Freedom Fighters, fighting alongside them and Sonic himself.
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 corp 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 advisors 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.
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 to shame.
Hell, her archery would make Hawkeye consider taking notes. And she was a traceuse (female parkour), an accomplished climber, and not a slouch with a sword either. She's one of the most martially adept princesses in Disney Canon, probably second only to Mulan.
In the Disney Aladdin cartoon, 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.
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 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 Mononoke doesn't control an actual kingdom, she does have her own family of wolves and fights pretty epically with a spear.
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 ThirdTake 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.
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 (failingly) 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. She is most assuredly badass.
Film - Live Action
Princess Leia of Star Wars 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 to take out the leader of the Ringwraiths. 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 Silmarillionis 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 in Dune, 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.
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 Calormen nobility (who later becomes Queen of Archeland).
Fisher from Hawk and Fisher by Simon Green is revealed to be a princess at one point. Which makes sense, as she'd started out as a deliberate Princess Classic subversion in Blue Moon Rising, before running off with the Prince Charming subversion to adopt new identities as Badass Cops.
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.
In Fiona Patton's Branion 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.
In A Song of Ice and Fire both Daenerys Targaryen and Arya Stark count. The latter only after her brother Robb Stark declares himself king.
King Arthur's daughter Rhianna in Katherine Roberts' Pendragon Legacy series.
The Xanth series: find one who's not i dare you, and that's includeing goblin cheif's daughters, married in, and other un-human princesses, really all female "main characters" are badass in their own rights.
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, which stuns Pocket when he finds out—because "she was rubbish at chess." Real war, he thinks, must be easier for her.
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 in Game of Thrones 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 though he could dump her and get away with that...
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.
Morgan also qualifies if she is the daughter of Say'ri or Lucina.
Midna, the titular Twilight Princess, definitely qualifies as well, given that she helps Link out in fighting enemies as well, paralyzing a group of enemies in an energy field, leaving them weak and defenseless while Wolf Link attacks them, and throwing a large Beast Ganon across the room, leaving him vulnerable.
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.
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; 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 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.
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.
Golden Sun: Dark Dawn has 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.
Touhou's setting of Gensokyo has an abnormally high level of princesses per capita, all of whom are quite skilled at danmaku and other forms of combat.
PC-98 exclusive Phantasmagoria of Dimension Dream had Kotohime, the aptly-themed "Maniacal Princess" who went head to head with a poltergeist, witches, a miko, Mima, a pallet-swapped doppelganger, and two scientific geniuses from another universe, armed only with some explosives and her smile. It's unclear whether her authority is In Name Only - in her ending she claims to be a police officer disguised as a princess - but it's implied that she did this all for the hell of it, and doesn't even have a wish to make when she beats the game.
Remilia Scarlet, Embodiment of Scarlet Devil's titular vampire, is an Ojou who claims to be a descendent of Dracula, and her theme is named "Septette for a Dead Princess." Word Of God is that she's lying and the track name is a misnomer, since there's no princess, no one's dead, and the theme is not even a septette. Nevertheless, Remilia is an endboss with the power to manipulate fate, while her little sister Flandre is an EX boss and a certified Person of Mass Destruction.
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.
Yggdra Yuril Artwaltz from Yggdra Union, the princess wielding a BFSdouble her size who strifes to defeat The Empire and liberates her home country.
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.
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.
Ame Lashiec, the main character of Heart Core, fits this trope preety well with her inhuman durability, magic powers and strengh. Being the daughter of 2 of the most powerfull demons in Asgard has its advantages.
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 Bad Ass. 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 powerfull magician and was capable of defeating most of the Overfiends singlehandedly.
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.
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.
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.
Adventure Time's Princess Bubblegum, without a doubt. She has the same voice actress as the aforementioned Starfire (except in the pilot). Coincidence?
And now we have the insanely powerful Flame Princess.
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 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.
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 by attacking him and his mooks 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.
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.
Eleanor of Aquitaine, Queen of England. You know you're badass when you can strong-arm the Pope into granting you a divorce from one king, marry another, and then have no less than three of your sons go on to be King.
One of those sons was Richard the Lionheart.
Another one was King John, but the third one was actually an OK king.
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 that succeeded in grabbing the crown by force. There also was one that 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 kingdom 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:
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.