In her third year at Hogwarts, Hermione is given a time-turner so she can take more classes. Several of her classes are concurrent to each other, so she attends one class (for 1-2 hours) goes back in time, and attends a different class during the same time frame.
But from an aging point of view, she should age 4 hours in relation to the 2 hours everyone else ages. How much older should Hermione be relative to everyone else after using the time-turner for an entire school year?
For those of you who aren't going to make it to the end of this post, it comes out to a week (plus half a day).
It is not certain whether wizards using time turners "age" faster, but for the sake of the question let's assume so and see if we can quantify somewhat.
According to the Harry Potter Wiki, there are 12 classes she could have been taking (7 core, 5 elective), and she signed up for all of them (Chamber of Secrets, first American edition, p.252). Assume she was allowed to take all of them (reasonable, I think). After dropping two classes (Divination and Muggle Studies), Hermione was able to resume a normal schedule. We'll take that as showing that ten courses fills the time allotted for classes completely. Let's say that's six hours a day, five days a week. That would make three hours per class per week; the two extra classes would then add six hours a week to Hermione's age. We'll assume she didn't use the TT for extra studying time on top of that (though I'm not quite certain of that).
The students arrived at school on September 1st as always, and Hermione dropped Divination around Easter. I didn't quickly find the event in the book, so we'll assume it was Easter (April 3, 1994). Now, if I counted right on my calendar, that's 29 weeks for Divination. The Christmas holidays remove two weeks from that, and we'll say another week for reading and exam period, so we'll say 26 weeks for three hours a week, making the added time from Divination three days and six hours.
She continued Muggle Studies through the end of the year. The wiki page says that's in the third week of June, for a total of 40 weeks (again assuming I counted correctly). Take out the three weeks off before Easter, and another two for reading, exams, and results, for a total of 35 weeks doubling for Muggle Studies another three hours a week. This comes out to four days and three hours.
Adding this to the Divination time and the three hours she and Harry spent saving Sirius and Buckbeak, we get seven days, twelve hours. Therefore, the answer appears to be just over a week.
+1 from me.
Hermione might also have reserved extra time for studying and additional sleep. So while the lower bound is, as Kevin says, 7.5 days, assuming she did not use her time turner over the holidays, she could have aged up to 32 hours per day. This potentially includes weekends (assuming that Hermione, often a stickler for rules, had permission to use the device so).
Assuming, then, there are 35 weeks of classes, we have an upper bound of 1470 hours, or about 60 days.
If, on the other hand, Hermione abused her time turner by using it during the holidays, we get as high as 70 days.
I have read some fan fiction that interpreted the restrictions on time turner usage as allowing one to go back six hours at a time, up to six times per day, for a total of an additional 36 hours per day. I think it's more straightforward to interpret each spin of the time turner as one use, for a total of six hours added per day. However, if this interpretation is correct, Hermione could have aged 460 days (and needed a second bed).
We are not told exactly how many classes she doubles up on, or how long they are, or how long the terms are, so a definitive answer is not really possible.
If it was one 2 hour class, 13 weeks a term, 2.5 terms, that would be 65 hours - 3 and a bit days. Even if these figures are out, it is liable to be a week older, not much more.
