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.