One of the items on the list of requirements is a copy of my rental agreement. I have no such thing. Do I just use some creative writing?

A rental agreement with signed copies of the owners house book ID card attached is a standard requirement at many immigration offices.

You should contact the owner and do a rental agreement (aka lease).

The place I have been staying these past years is owned by a Frenchman. Never had a lease. My girlfriend works at the villa as a caretaker and is provided an apt as part of her job. I stay there with her. It's the address I use for my 90 day report. 

Have the owner write a short statement that her accommodations is supplied by them and have her write one that you are staying with her. She should attach a copy of her ID card to it.