I have 2 cartons labeled with dthe ays of the week. I know what day each egg was laid. I've never had an egg last more than a week. If they get 6 days or so old I just give them to the neighbors. I suppose if I had more than a week's supply I'd wash a refrigerate. If you wash, you should refrigerate because removing the bloom makes the egg more susceptible to bacteria. Since refrigerating will shrink the egg and potentially draw surface bacteria through the shell I think it best to clean and sanitize of you are going to refrigerate.
Chickens will often sit on a nest with More than a week's worth of eggs. If they're sanitary enough to grow a healthy chick they're good enough to eat.