Looking this up for the US (the country where I live), it seems that only NH doesn't require car insurance (ninja edit: Virginia as well). So for me, this is an interesting hypothetical, but not in a way that's a part of my reality as I do not live in either of those two states.
>>At first thought, it appears more financially risky to drive!
As someone who does not own a car, and does not drive (and is poor - very poor in HN standards as well as in the city I live in), I would like drivers that I share the road with to have insurance, rather than drive without insurance. Getting hit by an uninsured driver would be an absolute nightmare that I doubt I could even afford to bring to court.
Not sure about the specifics of your car insurance but mine would cover that.
In /some/ places, car insurance is required by law. How does it work in places where it is not required by law? At first thought, it appears more financially risky to drive! I guess that insurance would be even more expensive because it is more likely that the other side is insured.