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Origami folding and thin structure buckling are intensively studied for structural transformations with large packing ratio for various biomedical, robotic, and aerospace applications. The folding of circular rings has shown bistable snap‐through deformation under simple twisting motion and demonstrates a large area change to 11% of its undeformed configuration. Motivated by the large area change and the self‐guided deformation through snap‐folding, it is intended to design ring origami assemblies with unprecedented packing ratios. Herein, through finite‐element analysis, snap‐folding behaviors of single ring with different geometries (circular, elliptical, rounded rectangular, and rounded triangular shapes) are studied for ring origami assemblies for functional foldable structures. Geometric parameters' effects on the foldability, stability, and the packing ratio are investigated and are validated experimentally. With different rings as basic building blocks, the folding of ring origami assemblies including linear‐patterned rounded rectangular rings, radial‐patterned elliptical rings, and 3D crossing circular rings is further experimentally demonstrated, which show significant packing ratios of 7% and 2.5% of the initial areas, and 0.3% of the initial volume, respectively. It is envisioned that the reported snap‐folding of origami rings will provide alternative strategies to design foldable/deployable structures and devices with reliable self‐guided deformation and large area change.