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This work uncovers a new class of criticality where eigenenergies and eigenstates of non-Hermitian lattice systems jump discontinuously across a critical point in the thermodynamic limit, unlike established Hermitian and non-Hermitian critical scenarios where spectrum remains continuous across a transition. Such critical behavior, dubbed the "critical skin effect", is rather generic, occuring whenever subsystems with dissimilar non-Hermitian skin localization lengths are coupled, however weakly. Due to the existence of this criticality, the thermodynamic limit and the zero-coupling limit cannot be exchanged, thus challenging the celebrated generalized Brillouin zone (GBZ) approach when applied to finite-size systems. As manifestations of the critical skin effect in finite-size systems, we present stimulating examples with anomalous scaling behavior regarding spectrum, correlation functions, entanglement entropy, and scale-free wavefunctions that decay exponentially rather than power-law. More spectacularly, topological in-gap modes can even be induced by changing the system size.