I'm Lucy, and I'd be happy to answer your questions today.
It's illegal to drive an unregistered car, or to drive without a license. The tow yard doesn't have any authority to arrest you - but there's no way to guarantee that they won't call and tell police that someone just drove away in an unregistered car, without a license. The tow yard also may refuse to release the car without proof of registration, if it was towed for not being registered and being on the public streets. The car could be traced back to you if the tow yard contacts the most recently registered owner about the tow fees, or if they pull a title to see who the current owner is. Then they could sue you for towing and storage fees - and that adds up. It could get very expensive, quickly.
You can transfer the title to someone else, have them register the car, and have them pick it up. In Missouri, a vehicle must be insured before it can be registered. If your intent is to have them eventually give the car back to you when you can drive it, then you should get something in writing that says so. The agreement will be enforceable as long as each of you receives something in exchange for the promise - for example, if the other party gets to drive the car for a specified amount of time, or something like that.
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