I (French citizen) am doing a PhD in computer science in the US: over the course of my PhD, how many days of internship am I allowed to do? I have a F1 visa. I plan to do several internships and would like to know the maximum amount of time I can work as an intern.
There's no maximum amount of time you can work as an intern. As long as you're complying with the F1 requirements (including being in good standing with your school and various degree-related and course-work related requirements) - you can work as many days as you wish.
However, there's a difference between full-time and part-time internships. Part time (up to 20 hours/week) does not affect your OPT. Anything more than that comes from your OPT quota (which is up to a year, so if you have a year or more full time CPT - you lost your OPT).
Here's a detailed explanation on the CPT (foreign students' internship) program from UPenn. Your school probably has a similar page, or an international students' office where you can inquire.