3 night-er near Steamboat

That would be the Zirkel Wilderness or the Flattops Wilderness hub. There are several TH's within a ~45 drive out of Steamboat.

The Zirkel's are the closest. I prefer the eastern TH's, but that's going to require a long drive. The topography and trail layout on the west slope of the Wilderness is going to limit you to just a long in/out. The Luna Lk trail with an extension to Roxy Ann "might" be long enough to fill 3 days, but it will still be an in/out hike through "Ground Zero" of the beetle epidemic that has decimated much of Colorado, and is believed to have started from the "Great Blow Down" in 1997.

The other option , a slightly further drive away would be the Flattops. There, you can cobble together several multi-day loops to fit your pleasure.

The "standard" loop is the circle around the Chinese Wall / Devil's Causeway from Stillwater Res. and can be done from either direction. Bear River Tr/ Chinese Wall Tr, W Lost Lakes Tr, Lost Lakes Tr..

If that's too short, you can extend it a good bit, but you're better off doing it CCW. Drop down the road to the Mandall Lakes TH and take it to Mandall Pass. The next 2 mi are xc but very easy. Drop down through the open tundra following Black Mountain Crk until you intersect the Black Mount Creek Tr, take that down until you hit the East Fork Tr and take that up until you intersect the Lost Lakes trail and follow the loop back CCW from there.

I have a hard time doing more than 5-7 mi day, in the Flattops, with so much good fishing to be had nearly every mile wherever you go. If all you want to do is roll up the miles,over 3 days, the long loop is a better option. If you're fisher "persons" the shorter loop might be better.

More loops are available from the Trappers Lake TH that will take you up on top of the major "Flattop Plateau", but you will constantly be at ~11,000' altitude

Just remember that mid August, is max Monsoon Season as well, and any high trail will be constantly exposed to lightening.

Also, in the same area is this

A little mis-information but I'll give these folks a pass due to old age. My preference is Flattops but it's 45 minutes to Yampa from Steamboat; another 45 minutes to the Stillwater TH. The pine beetle kill in Flattops and Zirkel is from an infestation that ended in the early 60's. There is no correlation to the latest infestation in the middle of the state. What was killed in the 60's blew down early in '97. Fires raged thru both wilderness areas at the same time in 2002 but the effects of both fires are long passed. I was bping in Flattops when the Big Fish fire blew up in 2002. The ranger left a note on my car to check in at the ranger station in Yampa if I wasn't dead. Also, CDT runs smack thru Zirkel. At the northern end the CDT trail is slightly west of Zirkel but the actual divide is right thru the heart. For the souther half CDT trail disects Zirkel.

Last year I took the Rainbow Lakes Trail into Zirkel then up to the CDT with Luna Lake as my turn around point. Camped at Upper Slide Lake my last night. Roxy Ann is just off this route. This is a nice out of the way hike with little traffic once by Rainbow Lake. From Stillwater in Flattops a good loop would be to Little Trapper then to Wall Lake and return with the Chinese wall. On this hike the devestation in the amphitheater at Trapper Lake is still fairly obvious.

BTW, I would never carry a bp over Devil's Causeway. I've been to both edges w a bp. I'd have no issues w a daypack. But, it's like the Grand Canyon...no way you can fall all the way to the bottom...I'd guess you'd only fall 1000 ft.

From the east side of the Zirkel's, Roxy Ann Lk might fit that itinerary, but there aren't many, if any, real "peaks" to bag.