A novel stable crystallographic structure is discovered in a variety of ABO3, ABF3 and A2O3 compounds (including materials of geological relevance, prototypes of multiferroics, exhibiting strong spin-orbit effects, etc...), via the use of first principles. This novel structure appears under hydrostatic pressure, and is the first post-post-perovskite phase to be found. It provides a successful solution to experimental puzzles in important systems, and is characterized by one-dimensional chains linked by group of two via edge-sharing oxygen/fluorine octahedra. Such unprecedented organization automatically results in anisotropic elastic properties and new magnetic arrangements. Depending on the system of choice, this post-post-perovskite structure also possesses electronic band gaps ranging from zero to ~ 10 eV being direct or indirect in nature, which emphasizes its universality and its potential to have striking, e.g., electrical or transport phenomena.