In 1967 Kraichnan predicted that for two-dimensional turbulence, the dependence of the energy on wavenumber should be given by E(k) ~ k^(-2/3). However, our experiments on nearly two-dimensional (low Rossby number) turbulent flows in a rapidly rotating annulus reveal instead E(k) ~ k^(-2). Measurements with and without a beta plane yield the same scaling. The probability distribution for velocity differences over a distance r exhibits broad non-Gaussian tails as a consequence of the coherent vortices. Surprisingly, the normalized distribution function is self-similar, i.e., independent of r. Our description of the observed vorticity distribution function exploits the underlying Hamiltonian structure of inviscid flows. The predicted distribution function is in good accord with the observations.