One of the most jaw-dropping acts in “Metamorphosis,” this year’s Big Apple Circus show, occurs when contortionist Odbayasakh Dorjoo fits her entire body inside an impossibly small transparent box.

Balancing on the palms of his hands, Odgerel Gombo folds his spine and bends his knees so that his toes rest comfortably on his shoulders. The contortionist from Ulaanbaatar, Mongolia, traveled and ...