We study the athermal mechanical response of deformable ring assemblies to quasistatic compression. Beyond jamming, further densification induces shape changes, resulting in macroscopic mechanical softening. Under cyclic compression, monodisperse systems anneal toward a nearly reversible path passing through an ordered state, whereas polydisperse systems converge to stable, hysteretic limit cycles. Our findings demonstrate how particle deformability governs collective self-organization and memory formation in jammed soft matter.