"Child of All Ages", a 1975 short story by P. J. Plauger, previously identified here as the answer to the question Trying to find a story about an immortal little girl. (There was a sequel called "The Con Artist" which I haven't read.) Any of these covers ring a bell?
The girl looks in her early teens in age (12-14). I think the story was told from the perspective of a man who meets her. She was either hitchhiking and he picks her up, or she is clearly a runaway who needs help (food or clothing or shelter from the weather) or maybe a prostitute he picks up on a street corner.
Melissa is passing for 14 but looks younger. She is not (currently) a hitchhiker, runaway, or prostitute. She is an orphan, living with foster parents. The beginning of the story is told from the perspective of a school psychologist, a woman, who is interviewing Melissa after she got suspended from school for arguing with her history teacher:
"My name is Melissa," she said, adding a nervous grin. "You must be Mrs. Foster." She was all little girl now, squirming the least little bit and kicking one shoe against another. The eyes shone with carefree youth.
May shook herself, slowly recovered. She thought she had seen everything before, until now. The guileless bit was perfect—Melissa looked more like a model eight-year-old than a chronic trouble-maker going on, what was it? Fourteen? Fourteen?
"You've been suspended from school for the third time this year, Melissa," she said with professional sternness. May turned on her best Authoritarian Glare, force three.
"Yep," the child said with no trace of contrition. The Glare faded, switched to Sympathetic Understanding.
"Do you want to tell me about it?" May asked softly.
"What's to say? Old man M—uh, Mr. Morrisey and I got into an argument again in history class." She giggled. "He had to pull rank on me to win." Straight face.
"Mr. Morrisey has been teaching history for many years," May placated. "Perhaps he felt that he knows more about the subject than you do."
"Morrisey has his head wedged!" May's eyebrows skyrocketed, but the girl ignored the reproach, in her irritation. "Do you know what he was trying to palm off on the class? He was trying to say that the Industrial Revolution in England was a step backward.
May and her husband agree to take Melissa in as her new foster parents. She tries a new tactic by telling them the truth about her long life.
So she starts telling him her story (I think she might be as much as 1000 years old.)
"Are you really twenty-four hundred years old?" asked George Foster, Sr. There, it was out in the open.
"Near as I can tell," spooning chicken and dumplings onto her plate. "Like I said, dates don't mean much to a kid. It was two or three hundred years before I gave much thought to when everything started. By then, it was a little hard to reconstruct. I make it twenty-four hundred and thirty-three years, now. Give or take a decade."
Because she is so young, she could never stay in one village or town for more than a few years, because people quickly realize she is not getting older.
"After my father died, I hung around Athens for a while—did I mention, that's where we lived? But too many people knew me and began to wonder out loud about why I wasn't growing up. Some of the other wizards began to eye me speculatively, before I wised up and got out of town. I didn't want to die a prisoner before anyone figured out I had nothing useful to divulge.
"I soon found out that I couldn't escape from my basic problem. There's always someone happy to take in a child, particularly a healthy one that's willing to do more than her share of the work. But after a few years, it would become obvious that I was not growing up like other children. Suspicion would lead to fear, and fear always leads to trouble. I've learned to judge to a nicety when it's time to move on."
As I recall she does not know why she is immortal, and has never meet any one else like her.
She knows why. Her father was a wizard:
"Not a magician, a wizard.." A little exasperated. "He didn't practice magic or cast spells; he was a wise man, a scholar. You could call him a scientist, except there wasn't too much science back then. Not that he didn't know a lot about some things—obviously he did—but he didn't work with an organized body of knowledge the way people do now."
[. . .]
"Anyway, he was working on a method of restoring youth. Everybody was, in those days. Very stylish. There was actually quite a bit of progress being made. I remember one old geezer actually renewed his sex life for about thirty years."
[. . .]
"My father spent most of his time trying to second-guess the competition. I suppose they were doing the same thing. His only real success was me. He found a way to stop the aging process just before puberty, and it's worked for me all this time."
In more recent times she may have earned money at times as a child prostitute. I think she may have asked him if he wanted to sleep with her, and when he reacts in shock, she tells him she is very experienced, and points out that at many times in the past in different cultures she would have been married at her apparent age, and had been.
Not exactly. Here's the part where she tells about her love life:
"Now I ask you, Grandpa, you tell me why I should want to grow up."
"There are other pleasures," he said eventually, "far deeper than the joys of childhood."
"You mean sex? Yes, I'm sure that's what you're referring to. Well, what makes you think I'm a virgin?"
He raised his arms in embarrassed protest, as if to ward such matters from his ears.
"No, wait a minute. You brought this up," she persisted, "look at me. Am I unattractive? Good teeth, no pock marks. No visible deformities. Why, a girl like me would make first-rate wife material in some circles. Particularly when the average life expectancy is, say, under thirty-five years—as it has been throughout much of history. Teen-age celibacy and late marriage are conceits that society has only recently come to afford."
She looked at him haughtily.
"I have had my share of lovers, and you can bet I've enjoyed them as much as they've enjoyed me. You don't need glands for that sort of thing so much as sensitive nerve endings—and a little understanding. Of course, my boyfriends were all a little disappointed when I failed to ripen up, but it was fun while it lasted.