A mechanically rugged form of stretchable electronics can be achieved through integration of functional materials and devices with composite substrates consisting of an ultralow modulus silicone adhesive layer on a strain-limiting fabric framework. The resulting system is sufficiently soft to enable extreme levels of deformation and non-invasive use on the skin, yet sufficiently robust for repetitive application/detachment. This letter introduces theoretical and experimental studies of mechanical designs, with optimization for a representative island-bridge device configuration to yield high levels of elastic stretchability. The physics of prestrain conversion and its role in enhancing the stretchability are systematically explored.