Physisorption or chemisorption from dilute polymer solutions often entails irreversible polymer-surface bonding. We present a theory of the resultant nonequilibrium layers. While the density profile and loop distribution are the same as for equilibrium layers, the final layer comprises a tightly bound inner part plus an outer part whose chains make only fN surface contacts where N is chain length. The contact fractions f follow a broad distribution, P(f) approximately f(-4/5), in rather close agreement with strong physisorption experiments [H. M. Schneider, Langmuir 12, 994 (1996)]].