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We find new effects for gravitational waves and memory in asymptotically-flat spacetimes of slow decay. In particular, we derive growing magnetic memory for these general systems. These effects do not arise in spacetimes resulting from data with fast decay towards infinity, including data that is stationary outside a compact set. The new results are derived for the Einstein vacuum as well as for the Einstein-fluid equations describing neutrino radiation, where the neutrino distribution falls off slowly towards infinity. Moreover, they hold for other matter and energy fields coupled to the Einstein equations as long as the data obey corresponding decay laws and other conditions are fulfilled. The magnetic memory occurs naturally in the Einstein vacuum regime of pure gravitation, and in the Einstein-matter systems satisfying the aforementioned conditions. As a main new effect, we find that there is diverging magnetic memory sourced by the magnetic part of the curvature. In the most extreme case, the magnetic memory, in addition, features a curl term from the neutrino cloud, growing at the same rate. Electric memory is diverging as well, sourced by the electric part of the curvature tensor and the corresponding energy-momentum component. Shear (news) adds to the electric memory. Moreover, a multitude of lower order terms contribute to both electric and magnetic memory. Further, we identify a range of decay rates for asymptotically-flat spacetimes for which the new effects occur but with different leading order behavior. The new effects are expected to be seen in current and future gravitational wave detectors. They have an abundance of applications of which we mention a few in this paper. Applications include exploring gravitational wave sources of the above types, detecting dark matter via gravitational waves and other areas of physics.