The effect of the coefficient of kurtosis, as a measure of third order nonlinearity, on the distribution of wave height maxima has been investigated. Measurements of the surface elevation during a storm at the North Alwyn platform in the North Sea have been used. The mean number of waves in the series is around 100. The maximum wave statistics have been compared with nonlinear theoretical distributions. It was found that the empirical probability densities of the maximum wave heights describe qualitatively the shift of the distribution modes toward higher values. The tendency for the peak of distribution to diminish with an increase in the coefficient of kurtosis up to 0.6 is also clearly seen. However, the empirical peak remains higher than the theoretically predicted one. The exceedance probability of the maximum wave heights was also estimated from the data and was compared with the theory. For the highest coefficients of kurtosis, estimated at nearly 0.6, the theoretical distribution approximates very well the empirical data. For lower coefficients of kurtosis, the theory tends to overestimate the exceedance probability of the maximum wave heights.