When I was in the 5th or 6th grade, I read this book about a princess who, well, does not want to be a princess. She sees her role as one big farce. So, she runs off into the mountains. There, she meets a dragon. The princess takes the dragon by surprise when she offers to take care of the cave where she and several other dragons reside. The dragon agrees and takes the young princess to her home and tells the dragons not to mess with the human. Word soon spreads of the princess's whereabouts, but the truth is twisted into a rumor that she was kidnapped. Now knights and princes are trying to "rescue" her in order to take her hand in marriage. I think there is another princess later in the book who meets the main heroine and stays with the dragons as well. Then they meet a prince who is cursed to be made of stone. The other princess and the stone prince end up falling in love. Does this story ring any bells for anybody?
The book / series you are thinking of is:
Dealing with Dragons by Patricia C. Wrede
Cimorene is everything a princess is not supposed to be: headstrong, tomboyish, smart - and bored. So bored that she runs away to live with a dragon - and finds the family and excitement she's been looking for.
There are four books in the Enchanted Forest Chronicles series.