I have read many FARs that describe the requirements for obtaining a pilot certificate but I have not found an FAR that states a pilot certificate is required in order to fly an aircraft - without passengers and not for compensation or hire. 14 CFR 61.3 details the requirements for a "required pilot flight crewmember" but I do not see that it states a person may not fly an aircraft unless that person has a pilot certificate.

What am I missing?