The existence of ultra-flexible low-energy forms of boron oxides (B2O3 and BO) is demonstrated, in particular structures in which B3O3 or B4O2 six-membered rings are linked by single B-O-B bridges. The minima in the energy landscapes are remarkably broad; the variation in the internal energies is very small over a very large range of volumes. Such volume changes may even exceed 200%. This remarkable behavior is attributed predominantly to the pronounced angular flexibility of the B-O-B bridges linking the rings, which is unusual for a covalent bond. At larger volumes, the structures are nanoporous; the pores collapse upon compression with negligible change in energy, making these suitable as guest-host materials. In marked contrast, in other materials where low density frameworks have been reported or predicted, such low-density phases are considerably higher in energy. The flexibility of the structures also offers a resolution of the long-standing controversy reconciling the structure and density of vitreous B2O3.