If I just bought a car and it's not registered and has no insurance, what would happen?

The owner of the car is jointly liable with the driver for the first $15,000 of your damages, with the driver then liable for any greater sum. Only if the driver had stolen the car would the owner not be liable. Whether your car was registered or you had insurance does not matter at all. The liability, however, is either the value of the vehicle or the cost of repair, which ever is lesser. It is difficult to convince an insurance company, but if you are stubborn enough, you might also get a few hundred more on the argument that if the car was new, its resale value has decreased as no one would pay as much for a car that had been in an accident than one never damaged.