Since we're talking about acetaminophen and ibuprofen, in France you're liable to end up with a medicine cabinet full of acetaminophen, because almost anytime you go to the doctor with any type of pain, they add it to the prescription.
(It is worthwhile to note, because Americans don't realize this, but not all medicines that a doctor prescribes for you is free, although the overwhelming majority will be.)
It depends on the insurance. I have pretty crap insurance and didn't expect them to cover a prescription for Omeprazole (aka Prilosec) since it's available OTC, but to my surprise they did.