Why would you not have a contract? Never agree to any work without setting out the terms before hand, such as how long the job will take and how much it will cost. Your best bet may be to contact a lawyer, but honestly without a contract there may not be anything you can do.
It may not be illegal but it sure is foolish not to have a contract with this company You are at their mercy as of now. That type of money being spent is really outrages. The remodel must be massive. I would let the contractor know it is time to finish this project and get it over with.
Technically, a check that's signed by you and deposited by the contractor would serve as a contract. What remains to be hammered out are the terms of that contract.
Did you get anything at all in writing? Like a bid, or a description of the scope of work? Those would be the terms of the contract, and they needn't be signed, as I stated before, you writing the check and he depositing it would show that both parties were in agreement.