<p>Exceptional points are non-Hermitian singularities associated with the coalescence of individual eigenvectors accompanied by the degeneracy of their complex energies. Since their discovery, exceptional points have attracted much interest and enabled numerous advanced applications, including sensing. However, accessing exceptional points generally requires delicate parameter tuning, and any related phenomena are intrinsically restricted to a very narrow bandwidth. Here we report a generalization of the concept of an exceptional point called an exceptional deficiency, which features the complete coalescence of entire eigenspaces with identical but arbitrarily large dimensions and the coincidence of entire spectral continua. We find that an exceptional deficiency can induce the anomalous absence or presence of the non-Hermitian skin effect, which transcends the established topological bulk–edge correspondence, resulting in unexpected synergistic skin-propagative dynamics. These phenomena are experimentally observed using active mechanical lattices. We further explore how exceptional deficiencies offer a route to the reliable and flexible control of localization and propagation and how they enable a high-sensitivity broadband sensor. The experimental demonstration of exceptional deficiency provides a new perspective on non-Hermitian physics and may impact related applications, such as sensing, modal control and lasing.</p>