Is it legal for a lawyer to take a retainer without a written agreement

This is not legal advice. Answer is found in the Ohio Rules of Professional Responsibility rule 1.5- "The nature and scope of the representation and the basis or rate of the fee and expenses for which the client will be responsible shall be communicated to the client, preferably in writing, before or within a reasonable time after commencing the representation, unless the lawyer will charge a client whom the lawyer has regularly represented on the same basis as previously charged."