Combining spatially resolved X-ray Laue diffraction with atomic-scale simulations, we observe how ion-irradiated tungsten undergoes a series of nonlinear structural transformations with increasing radiation exposure. Nanoscale defect-induced deformations accumulating above 0.02 displacements per atom (dpa) lead to highly fluctuating strains at ~0.1 dpa, collapsing into a driven quasisteady structural state above ~1 dpa. The driven asymptotic state is characterized by finely dispersed vacancy defects coexisting with an extended dislocation network and exhibits positive volumetric swelling, due to the creation of new crystallographic planes through self-interstitial coalescence, but negative lattice strain.