Changing the inherent physical capabilities of robots by metamorphosis has been a long-standing goal of engineers. However, this task is challenging because of physical constraints in the robot body, each component of which has a defined functionality. To date, self-reconfiguring robots have limitations in their on-site extensibility because of the large scale of today's unit modules and the complex administration of their coordination, which relies heavily on on-board electronic components. We present an approach to extending and changing the capabilities of a robot by enabling metamorphosis using self-folding origami "exoskeletons." We show how a cubical magnet "robot" can be remotely moved using a controllable magnetic field and hierarchically develop different morphologies by interfacing with different origami exoskeletons. Activated by heat, each exoskeleton is self-folded from a rectangular sheet, extending the capabilities of the initial robot, such as enabling the manipulation of objects or locomotion on the ground, water, or air. Activated by water, the exoskeletons can be removed and are interchangeable. Thus, the system represents an end-to-end (re)cycle. We also present several robot and exoskeleton designs, devices, and experiments with robot metamorphosis using exoskeletons.