Who would have thought that dirty hair could actually serve a purpose?
Maybe anyone who styled after skipping a shower (or two).
Most, if not all, of us were brought up to take baths or showers and wash our hair pretty much every day. That has changed. It's no longer the case, for whatever reason, but probably in part because slightly 'dirty' hair is often easier to style than freshly cleaned hair.
For instance, people with naturally dry and frizzy hair don't like to wash their hair as often, the reason being that our bodies, including our scalps, produce natural oils that are good for our hair.
Haven't you noticed after not washing your hair for a day or so that it becomes oily? Some people like to leave those oils in the hair, and by brushing it every day, those oils move down the hair shaft toward the ends and that is actually good for the hair. The oil gives the hair more flexibility and body, and it becomes less brittle than dry hair. Of course, really oily hair can become flat, boring and more difficult to style. We're looking for that happy spot in between the extremes.
To state the obvious, you can shower without washing your hair. Lots of us do that to protect an expensive new hairstyle before a special event.
People with naturally curly hair might not like it the day they shampoo because the curl becomes unruly right after a shower. After a day or two, the curl becomes far easier to control and style.
Another case in point, I always encourage my clients to not shampoo their hair on the day they want any kind of a style such as an updo. Hair that is not freshly washed holds better.
Another reason that some people skip washing hair is that they have extremely thick hair and don't want to take the time to wash and dry it every day.
So it's really up to you, according to what makes your hair look and feel the best. Everyone is different.