In 1935, explorer Eric Shipton had to abort an attempt up Mount Everest because he'd dawdled too long in the rhododendron-thick lower valleys and sprawling glaciers of a little-known Tibetan range to the east. Today, few people visit the Ama Drime mountains, leaving vast swaths of the area untrampled by Western soles. Spend 16 days exploring passes above 16,000 feet and camping on hillsides covered with giant blue poppies to reach the woodsy enclave of the Kharta Valley, east of Mount Everest. Count on walking about five to nine miles a day, stopping to chat with locals in remote nomad camps. Go easy on the homemade beer they'll offer; instead wander into the many side valleys of Qomolungma National Park. With numerous opportunities to bag 19,000-foot virgin summits, and natural hot springs to ease the pain, chances are you'll dawdle, too. OUTFITTER: High Asia; highasia.com PRICE: From $4,000 DIFFICULTY: Moderate to challenging WHEN TO GO: April–June, August–November