It is difficult to say. NHI has been changing the rules over the last couple of years due to budget issues.
There are some medicines that are not covered by NHI.
A private or semi-private room is not covered (you pay the difference).
I know that people do buy Cancer insurance, but I am not sure on whether that is through NHI, supplemental to NHI, or what.
Accidents should be covered by other insurance (like your car insurance in your example), but I am not sure of the details.
Not sure what she is saying about the operations, though. That doesn’t seem right.
These are some that I know. Personally, I don’t know of too many people who buy extra medical insurance, though. Usually, it seems to be life or income insurance.