We discuss and numerically test a method for direct and unambiguous measurement of ultrahigh laser intensities exceeding 1020W/cm2. The method is based on the use of multiple sequential tunneling ionization of heavy atoms with sufficiently high ionization potentials. We show that, due to a highly nonlinear dependence of tunneling ionization rates on the electromagnetic field strength, an offset in the charge distribution of ions appears sufficiently sensitive to the peak value of intensity in the laser focus. A simple analytic theory is presented which helps in estimating the maximal charge state produced at a given intensity via the tunnel-ionization mechanism. The theory also allows for calculating qualitatively a distribution in charge states generated in different parts of the laser focus. These qualitative predictions are supported by numerical simulations of the tunneling cascades developed in the interaction of a short intense laser pulse with a low-density target consisting of noble gases including argon, krypton, and xenon. Results of these simulations show that, using this technique, intensities in the range 1020-1024W/cm2 can be measured with sufficient reliability. The method could be extremely useful and of high demand in view of the expected commissioning of several new laser facilities capable of delivering ultrapowerful light pulses in this domain of intensities.