At what age can you get prescribed medication from a doctor on your own?

Depends. For what? In general, you need to be 18 to be able to give 'consent' = accepting a medication with an understanding of the risks and benefits. On occasion, doctors may prescribe medications for minor illness with what is called 'implied consent' -- previous written permission from parents to treat. Some ob-gyns may prescribe birth control pills to protect against teen pregnancy when parents disagree.