The time it takes a bound electron to respond to the electromagnetic force of light sets a fundamental speed limit on the dynamic control of matter and electromagnetic signal processing. Time-integrated measurements of the nonlinear refractive index(1) of matter indicate that the nonlinear response of bound electrons to optical fields is not instantaneous; however, a complete spectral characterization of the nonlinear susceptibility tensors(2)-which is essential to deduce the temporal response of a medium to arbitrary driving forces using spectral measurements-has not yet been achieved. With the establishment of attosecond chronoscopy(3-5), the impulsive response of positive-energy electrons to electromagnetic fields has been explored through ionization of atoms(6) and solids(7) by an extreme-ultraviolet attosecond pulse(8) or by strong near-infrared fields(9-11). However, none of the attosecond studies carried out so far have provided direct access to the nonlinear response of bound electrons. Here we demonstrate that intense optical attosecond pulses synthesized in the visible and nearby spectral ranges allow sub-femtosecond control and metrology of bound-electron dynamics. Vacuum ultraviolet spectra emanating from krypton atoms, exposed to intense waveform-controlled optical attosecond pulses, reveal a finite nonlinear response time of bound electrons of up to 115 attoseconds, which is sensitive to and controllable by the super-octave optical field. Our study could enable new spectroscopies of bound electrons in atomic, molecular or lattice potentials of solids(12), as well as light-based electronics operating on sub-femtosecond timescales and at petahertz rates(13-15).