Shape memory polymers are materials that can memorize temporary shapes and revert to their permanent shape upon exposure to an external stimulus such as heat1, light2, 3, moisture4 or magnetic field5. Such properties have enabled a variety of applications including deployable space structures6, biomedical devices7, 8, adaptive optical devices9, smart dry adhesives10 and fasteners11. The ultimate potential for a shape memory polymer, however, is limited by the number of temporary shapes it can memorize in each shape memory cycle and the ability to tune the shape memory transition temperature(s) for the targeted applications. Currently known shape memory polymers are capable of memorizing one or two temporary shapes, corresponding to dual- and triple-shape memory effects (also counting the permanent shape), respectively11, 12, 13. At the molecular level, the maximum number of temporary shapes a shape memory polymer can memorize correlates directly to the number of discrete reversible phase transitions (shape memory transitions) in the polymer11, 12, 13. Intuitively, one might deduce that multi-shape memory effects are achievable simply by introducing additional reversible phase transitions. The task of synthesizing a polymer with more than two distinctive and strongly bonded13 reversible phases, however, is extremely challenging. Tuning shape memory effects, on the other hand, is often achieved through tailoring the shape memory transition temperatures, which requires alteration in the material composition14, 15, 16. Here I show that the perfluorosulphonic acid ionomer (PFSA), which has only one broad reversible phase transition, exhibits dual-, triple-, and at least quadruple-shape memory effects, all highly tunable without any change to the material composition.