It's incredibly weather-dependent. I bicycled the width of Nevada over 5 days from Yosemite NP to St George UT, and was basically fine. Sweat evaporated immediately.

A couple years ago, my wife and I thru-hiked the Long Trail and got one (1) shower in 20 days. It was pretty rough despite availing ourselves of every opportunity to get a dip in a stream or pond (no soap). A day or two removed from a shower, we both felt pretty scuzzy. Everything got greasy and gross.

For context, we both thru-hiked the AT before we met; we are used to going without showers for days at a stretch. 10 was too many in the early summer humidity.