We know that both Bill and Percy Weasley got twelve OWLs. Probably Barty Crouch Jr. also got twelve OWLs, although Barty Crouch Sr. may have talked about something he wishfully imagined. There may have been more unknown students who got twelve OWLs, and there should also be students who failed some OWLs, but had originally started to study twelve subjects in their third year. Why, then, did Hermione Granger need a Time-Turner in Harry Potter and the Prisoner of Azkaban, if it wasn’t just for conveniently having time travel available when it was needed?
There are seven obligatory subjects: Astronomy, Charms, Configuration, Defence Against the Dark Arts, Herbology, History of Magic, and Potions (not counting Flying, because this is only for the first year). In Harry Potter and the Philosopher’s Stone, Chapter 14, Neville asks about Arithmancy and Ancient Runes, and Percy talks about Divination, Muggle Studies and Care of Magical Creatures, giving us a total of twelve subjects for the third year. Do we really know all optional subjects, or may there be at least one neither Percy nor Neville mentioned?
In Harry Potter and the Chamber of Secrets, Chapter 4, Hermione shows Harry and Ron the books for all optional subjects: Arithmancy, Care of Magical Creatures, Divination, Study of Ancient Runes, Muggle Studies – and there she is conveniently interrupted by Ron. This is such a common trope that it practically shouts “There’s another optional subject not mentioned yet! Prepare for future surprises. “
In Chapter 6 of the same book, Ron tries to discuss how Hermione’s classes for Divination, Muggle Studies and Arithmancy are at the same time on one day. Hermione reacts so annoyed that Ron never dares to discuss what else he might have seen on her timetable for other days. In Chapter 16, we see that Hermione has to sit double exams for Arithmancy and Transfiguration at Monday morning, and for Charms and Ancient Runes at Monday afternoon, but then Harry and Ron just give up and don't look at the other days, because there isn’t any point in asking how she’s going to do this. Is it possible that JKR deliberately avoided showing us full timetables because she knew that there had to be a sixth optional subject somewhere, but she had not yet decided what it would be?
We never see Hermione’s full timetables in Harry Potter and the Goblet of Fire either. Could JKR still not think of a plausible sixth optional subject, that wouldn’t be covered in other classes? Or did she forget that she had never actually said what it is?
In Harry Potter and the Order of the Phoenix, Chapter 31, at the last morning of the OWL examinations, Harry sits in the common room, reading through the notes he had borrowed from Hermione. After the Astronomy examination, Hagrid’s sacking and the aftermath at the previous evening, Ron is sleeping long. But wouldn’t sleeping long like Ron be out of character for Hermione? Shouldn’t she be right there, sharing her notes with Harry to prepare for the examination in History of Magic? Where is Hermione if she isn’t sitting another exam for the still mysterious sixth optional subject?
In Harry Potter and the Half-Blood Prince, Chapter 5, Harry, Ron and Hermione receive their OWL results. Hermione got ten "Outstandings" and one "Exceeds Expectations". Since she gave up Divination and Muggle Studies when she returned the Time-Turner, Hermione had tried to do something that had never been accomplished before: studying thirteen subjects during her third year.
So there is a good reason for giving Hermione a Time-Turner. No, it’s not for letting her try to get thirteen OWLs. It’s for teaching her these important lessons:
Magic cannot do everything.
Your Heart’s Desire may always be out of reach.
Even if you can add some extra hours to your day, it’s still impossible to learn everything that can be learned.
Addendum 1:
Uh, what? A newer edition of Harry Potter and the Half-Blood Prince says that Hermione got only nine "Outstandings"? Did somebody complain that eleven OWLs for Hermione is WRONG and should be edited, because this would require her to have started studying thirteen subjects while there are only twelve subjects available?
Fanon says that time travel isn’t really such a big secret and everybody who studies more than ten subjects at Hogwarts gets a Time-Turner. This theory is very popular because people just love to use time travel in their fan fiction. JKR tried to shoot it down on Pottermore, saying that time travel is really very dangerous and really top secret, and that Hermione was really the only student who ever used a Time-Turner at Hogwarts, but nobody believes her. That’s what you get when you listen to your readers and throw years of planning and deliberately writing around the mysterious subject out of the window because after so many years you don’t remember why the plot of your second book requires Hermione to get eleven OWLs in the sixth book.
Unfortunately the Department of Inaccuracies, even if it existed, can’t do anything about this.
Addendum 2:
Look, Ron, it isn’t too difficult. It took me only ten years, and I didn’t even lurk on Hermione’s timetables. BICIMTG is "Basics of Interspecies Communication: Introduction to Mermish, Troll and Gobbledegook".
It’s sufficiently magic, not covered in other classes, not requiring an additional teacher we have never seen, and Hermione never talked about it because she was just too curious whether Ron would ever figure it out on his own. (Thanks to Julia H. of the Harry Potter Lexicon Forum for suggesting that Professor Dumbledore could teach Mermish and probably also other languages.)
HG