The best candidate in my mind is Balsa from
Moribito. She is a thirty-year-old spear-wielding bodyguard who is tasked with protecting a young prince from assassination. Balsa is one of the finest women in anime with a strong moral character and an elaborate back story.
A younger contender is Erin, of
Kemono no Sou-ja Erin, based on another novel by the author of the
Moribito series. Erin wishes to follow her mother and become a veterinarian to mystical beasts. Later she discovers she possesses an unusual ability to communicate with them and becomes embroiled in the center of a political struggle. We see Erin develop from about age eight to eighteen. (I can't link to the fansubs for
Erin because it is licensed, but they have a superior translation to Crunchyroll's efforts and are worth searching for.)
Another strong teen-aged female is Shion, the heroine of
Shion no Ou. She witnesses her parents' brutal murder at age four and becomes mute in response. We pick up her story at age thirteen when she is contending to become a professional Shougi ("Japanese chess") player like her adoptive father. I found Shion's rise through the ranks of Shougi players more interesting than the solving of the murder mystery which runs in parallel to her story.
I'll also recommend Nina Fortner, the main female character in
Monster. Though Nina gets less screen time than the main lead, Dr. Tenma, she is a brave and intelligent young woman and definitely worthy of admiration.
Another excellent choice would be Kou Shurrei who is the star of
Saiunkoku Monogatari. Shurrei desires to become the first female official of her country (modelled on T'ang China). The problem you'll face is finding a copy to watch. It went out-of-print a couple years back and only torrents remain. They are a bit hard to find, though.
Finally I'll suggest Nakajima Youko of
Junni Kokki ("Twelve Kingdoms" in English). At the outset Youko is a timid high-school girl who is suddenly whisked off to the world of the Kingdoms where she discovers her true identity and role in life. Her growth as a character is unmatched in anime that I've seen. By the end she is truly an inspiring character, as are the young ladies who join her.