State laws vary so you will need to check with your state's Department of Motor Vehicle agency, or similar entity, in your specific state. They should have a list of vehicles that are exempt from your state's mandatory insurance requirement laws.
For example, in Washington state there are vehicles, such as motorcycles and mopeds, that are exempt from the mandatory insurance requirements but they are still subject to the financial responsibility law. This law requires the person responsible for damage to another person's property or person to pay for damages they caused.
In Oregon there are certain vehicles, such as antique motor vehicles, farm trailers, farm tractors, motor assisted scooters, mobility devices, snowmobiles and all-terrain vehicles, which are exempt from the mandatory insurance requirements.
The State of Georgia allows for non-motorized vehicles, such as trailers and campers to be exempt from registration while snowmobiles, bicycles and golf carts are a few vehicles that are not required to carry insurance coverage.
Remember if you take the vehicle that is exempt from insurance in your state to a different state to check their laws since mandatory insurance laws do differ between states. If you find out your certain vehicle does need insurance get your car and insurance quote here or your motorcycle insurance quote here.