Magnetoelectric coupling-the ability of a material to magnetize upon application of an electric field and, conversely, to polarize under the action of a magnetic field-is rare and restricted to a rather small set of exotic hard crystalline materials. Intense research activity has recently ensued on materials development, fundamental scientific issues, and applications related to this phenomenon. This tantalizing property, if present in adequate strength at room temperature, can be used to pave the way for next-generation memory devices such as miniature magnetic random access memories and multiple state memory bits, sensors, energy harvesting, spintronics, among others. In this Rapid Communication, we prove the existence of an overlooked strain mediated nonlinear mechanism that can be used to universally induce the giant magnetoelectric effect in all (sufficiently) soft dielectric materials. For soft polymer foams-which, for instance, may be used in stretchable electronics-we predict room-temperature magnetoelectric coefficients that are comparable to the best known (hard) composite materials created. We also argue, based on a simple quantitative model, that magnetoreception in some biological contexts (e.g., birds) most likely utilizes this very mechanism.