I have a daughter. She goes to pre-school. At pre-school, the girls play princess and the boys play superhero. (And vice versa, sometimes, but mostly not.)
So my daughter is crazy for princesses. And, to a lesser extent, superheroes.
I, on the other hand, gag every time I see a Disney princess. I see "Little Mermaid", I think "kills herself for love". I'm OK with Cinderella in general, not so much with the blond chick in the fluffy pink dress.
Net result? We do princesses. We just don't do Disney princesses. Or their ilk. We do princesses who are fundamentally superheroes with fewer tights and more crowns. They also often wear capes, which interestingly are common ground between princess outfits and superhero outfits.
Here are some personal recommendations for princess books we've enjoyed. (Note that this is not an exhaustive list; it has to be books I've read and liked.)
See also "boy" books with strong female characters
Princess Books for the Picture Book Set
Starred books include dragons.
*The Paper Bag Princess The dragon abducts the prince, the princess rescues him, by being brave and clever. Also, they do not get married.
Anything by Jay Williams with a princess in it, notably *The Practical Princess (note that there are two books by this name, a picture book and an almost un-illustrated compilation of a bunch of his stories) in which the princess kills the dragon, Petronella (there are two of these, too, but they differ only in illustrations), Stupid Marco
The Princess and the Pizza The princess does not end up marrying the prince, but does end up inventing the pizza and saving her family from poverty.
Sleeping Ugly The princess is never woken up; the prince marries somebody else instead. Being a princess does not entitle you to be rude.
Falling for Rapunzel This is a very silly book in which the prince does not end up marrying the princess.
The Princess Knight The princess secretly learns to ride and use weapons, and fights in the tournament that's going to determine who gets to marry the princess. As a bonus, the princess also learns to fight and controls her horse through gentle means.
Elisa Kleven's The Paper Princess As the title suggests, this is about a paper doll, not a real princess. It's a beautiful, sweet book, with a good moral (and two equally lovely sequels) but literalists may object.
Robert San Souci's The Well at the End of the World The princess saves the day by being smart and nice but not beautiful. It's a little heavy-handed on the beauty thing, actually, but the heroine princess, who is particularly good at accounting and doesn't care that she's not beautiful, is a great character.
Princess Books for Elementary School
*Laurence Yep's Dragon Series In this case, the dragon IS the princess. (She is capable of taking human form.) There is also a human boy as a hero. These are based in Chinese mythology.
*The Enchanted Forest Chronicles The princess and the dragon are in league (not all princesses are brave and not all dragons are good in these stories, but the main characters are a brave princess and and a good dragon).
*Bone This graphic novel series has a dragon from the beginning, but you don't get to the princess for a while. Still, although the title character is apparently male, all the strongest characters are women. You've got a fighting princess, a fighting grandmother... The dragon is on the same side as the princess.
The Serpent Slayer Not all princesses, but a bunch of strong women from various cultures.
Castle In The Air The main character is a carpet-seller. However, a huge variety of princesses do enter into it heroically.
*The Courageous Princess The princess saves herself, and an entire country, from a dragon. This is a graphic novel, with lovely pictures.
Gail Carson Levine's Princess Tales series We have three of these (which are more-or-less novella length). The Fairy's Mistake and The Princess Test are both interesting twists on traditional fairy tales, which involve strong girls who are not princesses by birth. (These are not particularly royalist princess stories.) *The Two Princesses of Bamarre is my favorite. The moralizing about being brave is a trifle heavy handed, but hey, it's a fairy tale, they're not supposed to be subtle. What I like a lot is that the dragon, who is definitely a villain, is depicted as an abuser. Under the up-front moral there's this quieter one: somebody who hurts you and then says you made them do it and they love you is crazy and dangerous.
Pamela Dean's Secret Country series These are a "kids fantasy universe become true" story. They work in a lot of moral complexity. Life as a princess is not everything one might have hoped for; it's exciting but also difficult, scary, confusing...
Kate DiCamilo's The Tale of Despereaux This is only marginally a story about a princess. It's a story about other characters (the majority of them talking rodents) with a princess in it. However, she does save the day, if only by being really, really nice. It's a gripping read and much more complicated than it seems like it's going to be (when, for instance, the title character falls in love with the princess for no reason at all). Nonetheless I'm not sure I actually like it. I'm not about to forget it, though.
*Anna Quindlen's Happily Ever After Another (younger) version of "wishes to be princess, finds out it's not all that great". Plus "princess in danger rescues herself, finds out princes not that great at rescuing, not everybody is as evil as they're supposed to be." Extra points for best use of chamberpot. Hey, just mentioning the existence of chamberpots ought to be good for extra points.
Margaret Haddix' Just Ella Technically, this book does not have a princess in it. Cinderella is a commoner until she marries the prince - and in this version, she doesn't end up marrying the prince. But this book starts out with her living in the palace, after the ball, learning to be a princess. And discovering that she HATES everything about it. So she does something about it. (She does not, however, succeed in quashing the ridiculous and annoying rumours that she had a fairy godmother.)
Garth Nix' One Beastly Beast This has four stories (novellas?), two with boys as heroes, two with girls. Opal's favorite, though is the one about the princess and the Beastly Beast. (Her second favorite is the other one about a girl. Coincidence? Probably not, as I've seen this pattern in other books.)
The Pirate Princess series These are not great literature, but hey! Princesses! Who are pirates! Swashbuckling, derring-do, cardboard villains, talking parrots. These are easy readers, and Opal adores them totally. They are surprisingly hard to come by in the US (they're UK books, available in Australian airports, even), but can be found used at alibris or Amazon and sometimes at Book Closeouts
Shannon Hale's Rapunzel's Revenge Another graphic novel. Rapunzel retold in a very fictional wild west. Rapunzel saves herself, and a lot of other people. There is an all-too predictable romance, but there are a whole bunch of women in positions of power, some of them good and some of them bad, and there is a lot of racial diversity.
Princess Books for People Who Can Handle More Complexity
The Goose Girl by Shannon Hale Here's a princess who does eventually marry the handsome prince she's supposed to marry, but only after she goes through horrible tribulations and averts a war.
Megan Whalen Turner's Queen's Thief trilogy These are theoretically young adult fiction. They contain several princesses, most of whom become queens, and none of whom you would want to mess with. They are genuine rulers, in a time of war, who make hard decisions. And kill people. Most of whom deserve it.
The Princess Bride The book is definitely related to the movie, but it's different. The princess is better in the book, for one thing. But they're both comic swashbuckling romps.
*Robin McKinley's The Hero and the Crown
Dragons are very definitely bad guys here, and the princess is a dragon-killer. A complicated story with a simple ending, but good characters.
Playing Princess
We make our own dress-up clothes; a cape and a crown are enough to make a princess, but the occasional fluffy skirt doesn't hurt.
For dolls, check out Playmobil knights. They're not intended to be princesses, but a lot of them have long hair and occasionally some other girly quality (like a pink shield, or a swan on a helmet).