We perform an ab initio and accurate exploration of the subcycle transient multiphoton-ionization dynamics of atomic and molecular systems subject to intense near-infrared laser fields on the subfemtosecond time scale. Multiple-ionization bursts within a single optical cycle are found in the time-dependent ionization rates not only for diatomic molecules H2+ and HHe2+, but also for the hydrogen atom. The analysis of the electron density reveals that several distinct density portions can be shaped and detached from the target within a half cycle of the laser field.