Depends on where and what...
I can only tell you about Germany.
Here it would be illegal to have a car without a valid third party liability insurance. It would even be impossible to get a license plate for a new car or to change ownership of a car without an attest that you have an insurance.
In Germany it is illegal for any employer to employ a person who does not have health insurance, pension insurance, unemployment-insurance and nursing-care dependency insurance. Also the employer will have to pay half of these insurances. People who are unemployed and depend on social security payments (yep that's where the general unemployment insurance pays off) are automatically health-insured too.
It is also near impossible to have any business without third-person liability insurance.
Then, if you own any real estate property in Germany, you need third-person liability insurance. Also, if you want to buy real estate property on credit, you will not get a credit from any bank without proper insurance of the value.
Where I'm from, taking out health or car insurance is not compulsory. Most people who drive bombs (old cars) tend to only take out third party insurance. That is, if they are involved in a car crash and it is their fault, their insurance will pay for the other car to be repaired but they wouldn't get their own car repaired. If someone is not insured and the other person is, their insurance company will take the uninsured person to court if the insured person is at fault.. Health insurance in Australia is not compulsory, if you don't take out health insurance you will be taxed more but our public system is free so if you don't have insurance cover, you will still get treated by doctors in a public hospital. The only disadvantage for not having health insurance is if the injury is not life threatening you will be placed on a waiting list (the wait could be 2 to 10 years before your injury is treated), so people who can afford health insurance tend to take it.
Depends on where you live. In most developing countries (poor countries too), insurance (any kind of insurance) is not something important or even considered exist since the citizens are too busy to feed their families. In US or in other developed countries, I think it is.
Car insurance? Yez it is. U can get a ticket. I know i gotta pay that next week for letting it lapse. Got pulled over. Health insurance no