More than 1,700 people vie for 62 private launch permits to run the Upper Selway in the yearly U.S. Forest Service lottery. The prize? The chance to have one of the most spectacular rivers in America almost all to yourself. Paddle 47 miles of snowmelt-engorged Class III and IV rapids cutting through the 1.3-millionacre Selway-Bitterroot Wilderness, along the Idaho-Montana state line, passing steep granite hillsides thick with cedar and fir trees. Go for whitewater in early summer, hungry cutthroat trout in July. Bypass the lottery by reserving a year in advance with an outfitter like Three Rivers Rafting (five-day trip, $1,800 per person; idaho3rivers.com).