Cuzco and Machu Picchu
Cuzco sits at 11,200' and is one of the very few cities in the world where restaurants keep oxygen tanks at the ready for when patrons pass out while eating dinner. (We saw it happen. There's also a free oxygen dispenser in the arrivals area of the airport.) Its main attraction is the Inca ruin Sacsayhuaman (the mnemonic is "sexy woman"), built of impossibly large stones that fit together impossibly well.
Cuzco is a 3-hour train ride from there to Machu Picchu through the Urubamba river valley. Machu Picchu itself sits in the peaks above the village of Aguas Caliente and, quite simply, has to be seen to be believed. No pictures can possibly do it justice. (But we took an awful lot of them anyway.) In Aguas Caliente we stayed at the beautiful Inkaterra hotel, nestled into the rainforest at the base of the steep mountainside, right on the river.