Princess Charlotte in the Berserk anime, and the origin arc of the manga, is the sheltered and demure version of this. After the Eclipse, the Kushan empire took Wyndham and Charlotte was taken prisoner by Ganishka. Griffith rescues her after being reincarnated back on earth. She's now part of his new Band of the Hawk.
Of the "core" Disney Princesses, the three earliest, Snow White, Cinderella and Aurora, fit this trope (even though Cinderella is a princess by marriage). Others like Ariel, Belle and Jasmine have stories that are a bit more complex.
One half of the soul and fiber of Shrek is about subverting this one by turning Princess Fiona into The Lad-ette. It is deconstructed further in Shrek the Third. Sleeping Beauty is narcoleptic, Cinderella is an obsessive-compulsive cleaner, Snow White summons woodland creatures with Led Zeppelin, and Rapunzel wears a wig and is also The Mole.
Sara Crewe in the movie versions of A Little Princess is about as close as this trope gets without any actual royalty. But in the book she isn't sweetness and light. Part of why she is disliked is because of her ability to make adults uncomfortable by standing up for herself. Unfortunately, the Shirley Temple version completely destroys what made the character great, while adding in an inability to face reality. Whereas in the book she is remarkably competent and tough for her age. The 90s version seems to fix that though.
Dany is also a deconstruction; she starts out an impoverished Princess Classic, but then loses a great deal of her innocence (including her virginity) when she's sold as a wife to Khal Drogo, swaps the violet silks for painted leather, learns to ride and rule like a Mongol warlord, loses her husband, has the woman who betrayed her burnt alive, hatches some dragons, creates a new army all on her own, handles multiple assassination attempts, faces down a house full of warlocks, gains a brand new army, burns down entire cities full of slavers, and eventually sets up shop with an army of freed slaves and starts ruling a city as an imposing empress, Daenerys Stormborn, Mother of Dragons.
In Xanth, Princess Irene was a Royal Brat. Her daughters, Ivy and Ida are straight examples.
The Lady Amalthea in The Last Unicorn acts like this as she loses her memories of being the last unicorn.
Averted in Mercedes Lackey's Valdemar books by Princess Elspeth, who starts off a Royal Brat and becomes a tomboy after she's broken of the brattishness. Eventually she abdicates her position as heir when it becomes obvious (to her, at least) that she'll serve her country better as a Herald-Mage than as its queen.
Poked fun at in Angry Lead Skies, when Garrett ponders Kip Prose's childish fantasies about saving beautiful princesses, and how, despite having run into everything else on his weird cases, he's never met a Princess Classic. Near the end, Garrett does catch sight of two genuine royal daughters, and they're both quite unattractive, yet are praised and fawned upon constantly by political kiss-ups.
Firebird Trilogy: Princess Carradee Angelo is gentle, honorable, sweet, and rather uninterested in politics. Unfortunately for her, as eldest, she is the heir to the throne, which leads to her being thrust into ruling before she is truly ready for it. She contrasts with her sisters Phoena, who is a Politically-Active Princess, and Firebird, who is a Badass Princess.
Used in the most famous Halloween episode of Buffy the Vampire Slayer. Buffy dresses up as one of these on the Halloween when Ethan Rayne conjures up a spell that causes her and all of her friends to get turned into their costumes. So Willow, who dressed as a ghost, becomes a ghost and Xander, who dressed as an Army Guy, becomes an actual Army Guy. Ethan lucked out when she chose the Princess costume: the powerful Slayer becomes a simpering, helpless bundle of nerves and snobbery.
Played with on Merlin with the portrayal of Princess Mithian. Characterized as the embodiment of a Princess Classic, she's genuinely lovely, as well as beautiful, royal, elegant, talented and with an extensive wardrobe of gorgeous outfits; yet it's all done in order to provide more impact to the fact that King Arthur turns her down for the sake of the blacksmith's daughter.
Fairytale imagery featured on Taylor Swift's second album, Fearless. She explored the disconnect "between fairy tales and the reality of love": "We're raised as little girls to think that we're a princess and that Prince Charming is going to sweep us off our feet". Seen in "Love Story", "White Horse", and "Today Was A Fairy-Tale".
Princess Shine from Super Robot Wars often struggles to maintain the image, particularly in her language, and of course, the fact that most Princesses don't take on threats to her Kingdom head on in a (oddly cute) Humongous Mecha.
Debatable if there are actual examples in Real Life (that weren't this from being sheltered and not taught about the world), but Queen Victoria of England did her best to make royal ladies appear to be that way. To the point of the Urban Legend of noble women being advised to "Close your eyes and think of England".
On the subject of Saints with a royal title, Saint Catherine of Alexandria provides a very early example of this trope (her name even comes from "Katharos", the Greek word meaning "pure"!. It is a somewhat unusual one, however, as the virgin princess is wed to Jesus Christ in mystic marriage (ie. a dream-vision) and chooses martyrdom rather than betraying her vows to Him!