The ionization dynamics of small rare-gas clusters in intense, ultrafast laser fields are studied via classical trajectory Monte Carlo simulations. Our results indicate that for similar laser pulses the charge states reached by atoms in a cluster can be significantly higher than those for atoms in the gas phase. The ionization enhancement is strongly dependent on the cluster density and exhibits a rapid increase in charge state once the laser intensity has reached the threshold for single ionization. This {open_quotes}ionization ignition model{close_quotes} is driven by the combination of the laser field and the strong field from the ionized cluster atoms. Approximate atomic inner-shell ionization probabilities are calculated for several cluster densities and peak laser intensities and provide evidence for the generation of inner-shell holes on an ultrafast time scale. This is a necessary condition for the generation of x-ray pulses with temporal widths comparable to that of the driving laser pulses. {copyright} {ital 1997} {ital The American Physical Society}