Bar-headed geese are renowned for migratory flights at extremely high altitudes over the world's tallest mountains, the Himalayas, where partial pressure of oxygen is dramatically reduced while flight costs, in terms of rate of oxygen consumption, are greatly increased. Such a mismatch is paradoxical, and it is not clear why geese might fly higher than is absolutely necessary. In addition, direct empirical measurements of high-altitude flight are lacking. We test whether migrating bar-headed geese actually minimize flight altitude and make use of favourable winds to reduce flight costs. By tracking 91 geese, we show that these birds typically travel through the valleys of the Himalayas and not over the summits. We report maximum flight altitudes of 7290 m and 6540 m for southbound and northbound geese, respectively, but with 95 per cent of locations received from less than 5489 m. Geese travelled along a route that was 112 km longer than the great circle (shortest distance) route, with transit ground speeds suggesting that they rarely profited from tailwinds. Bar-headed geese from these eastern populations generally travel only as high as the terrain beneath them dictates and rarely in profitable winds. Nevertheless, their migration represents an enormous challenge in conditions where humans and other mammals are only able to operate at levels well below their sea-level maxima.