First we'll tackle the one we can't insure in New Zealand, emergency medicine. In New Zealand we don't have private emergency care, though it is often asked for.
This is possibly due to the American TV influence, where private emergency care is common.
If you are in need of emergency medical care, hang up the phone, close the web browser and call 111. St John's ambulance and the public hospital system is your only choice.
Once you are stable, then medical insurance comes into it's own.
Next - GP & Prescription costs