that would be near impossible. You need to have your business registered.. you could keep your performance records hidden good or bad. But there needs to be a name associated with said company even if it's a shell company..there are tracks..giving enough time one can follow.
CEOs are supposed to be very outward...the job entails meeting people and making connections...if you wanted Privacy your better off being a COO. Even if it's ceremonial role and you don't own the company... your name is Associated with it's business and all over the place. Even if it's a nickname theres usually tracks to find it . getting contact with said person is a bit tricker since they hide numbers, Scan calls and Fake emails for obvious security reasons... owner is not necessarily the CEO.
unless your a small company that involves you and your family... you still have to register as a business to operate. Even companies most of us probably never heard of.
How would pedro with company of 4 people claim to be a CEO of a "decent" business that isn't registered, apply to be a president, and refuse to give a resume?