Well actually that one is pretty easy. If you are European you can ask for your European Health Insurance Card (free, valid for several years) which will cover you in the whole EEA as if you were a resident yourself.

The problem though is that if you have a chronic or permanent disease that requires continuous medication. Hospital will likely sort you out in a pinch, but if you need to stock up some pills you will need to go through the whole doctor chain (GP / Specialist / Various treatment trials ...) in each country (prescription do not cross borders). I have known people going back to their country on a regular basis just because that´s just a pain in the ass to go back to square one and start a several months ordeal only to end up with the medication you know is working.