GitHub Education provides free private repositories for educators.

You can check it out here: https://education.github.com/discount_requests/new

As I recall, that's only for use in the classroom, to get students hooked. If you're doing research, or anything other than teaching a class, you pay for private repos. Feel free to correct me if I'm wrong.

Bitbucket gives you a free unlimited account if you sign up with a .edu email address.

[Edit: You actually get five free repos. Still limited and not at all the deal you get with Bitbucket.]

No, GitHub gives you 5 free private repositories for any educational use if you sign up with an academic email. I use mine for research.

Researchers are eligible too.