The escape transition of a polymer "mushroom" (a flexible chain grafted to a flat nonadsorbing substrate surface in a good solvent) occurs when the polymer is compressed by a cylindrical piston of radius R, that by far exceeds the chain gyration radius. At this transition, the chain conformation abruptly changes from a two-dimensional self-avoiding walk of blobs (of diameter H, the height of the piston above the substrate) to a "flower conformation", i.e., stretched almost one-dimensional string of blobs (with end-to-end distance R) and an "escaped" part of the chain, the "crown", outside the piston. The extension of this problem to the case of star polymers with f arms is considered, assuming that the center of the star is grafted to the substrate. The question is considered whether under compression the arms escape all together or whether there occurs an arm by arm escape under increasing compression. Both self-consistent field calculations and molecular dynamics simulations are found to favor the latter scenario.