We explore the nonlinear dynamics of a driven power-law oscillator whose shape varies periodically in time covering a broad spectrum of anharmonicities. Combining weak and strong confinement of different geometry within a single driving period, the phase space allows not only for regular and chaotic bounded motion but in particular also for an unbounded motion which exhibits an exponential net growth of the corresponding energies. Our computational study shows that phases of motion with energy gain and loss as well as approximate energy conservation alternate within a single period of the oscillator and can be assigned to the change of the underlying confinement geometry. We demonstrate how the crossover from a single- to a two-component phase space takes place with varying frequency and amplitude and analyze the corresponding volumes in phase space. In the high-frequency regime an effective potential is derived that combines the different features of the driven power-law oscillator. Possible experimental realizations are discussed.