A combination of experiment and theory shows that dielectric elastomers exhibit complex interplay of nonlinear processes. Membranes of a dielectric elastomer are prepared in various states of prestretches by using rigid clamps and mechanical forces. Upon actuation by voltage, some membranes form wrinkles followed by snap-through instability, others form wrinkles without the snap-through instability, and still others fail by local instability without forming wrinkles. Membranes surviving these nonlinear processes are found to attain a constant dielectric strength, independent of the state of prestretches. Giant voltage-induced stretch of 3.6 is attained.