Great question.  Have you checked with your girlfriend's insurance to see if her coverage includes other drivers? According to Ohio Bureau of Motor Vehicles, it is illegal to drive in Ohio without insurance or other financial responsibility (FR) proof.  In fact, it is illegal for any motor vehicle owner to let someone else drive the car without FR proof.

Section 4509.101 of the Ohio Revised Code prohibits a individual from operating a motor vehicle in Ohio without maintaining proof of FR continuously throughout the registration period with respect to that vehicle, or, in the case of a driver who is not the owner, with respect to that driver's operation of that vehicle.  The law requires financial responsibility in the minimum amount of $12,500 for bodily injury to or death of one (1) individual in any one (1) accident, $25,000 for bodily injury to or death of two (2) or more individuals in any one (1) accident, and $7,500 for injury to the property of others in any one (1) accident.