Strong-field laser excitation of solids can produce extremely nonlinear electronic and optical behaviour. As recently demonstrated, this includes the generation of high harmonics extending into the vacuum-ultraviolet and extreme-ultraviolet regions of the electromagnetic spectrum1,2,3,4,5,6,7,8. High harmonic generation is shown to occur fundamentally differently in solids and in dilute atomic gases1,2,3,4,5,6,9,10,11,12,13. How the microscopic mechanisms in the solid and the gas differ remains a topic of intense debate1,2,3,4,5,6,7,8,9,10,11,14,15,16,17,18. Here we report a direct comparison of high harmonic generation in the solid and gas phases of argon and krypton. Owing to the weak van der Waals interaction, rare (noble)-gas solids are a near-ideal medium in which to study the role of high density and periodicity in the generation process. We find that the high harmonic generation spectra from the rare-gas solids exhibit multiple plateaus extending well beyond the atomic limit of the corresponding gas-phase harmonics measured under similar conditions. The appearance of multiple plateaus indicates strong interband couplings involving multiple single-particle bands. We also compare the dependence of the solid and gas harmonic yield on laser ellipticity and find that they are similar, suggesting the importance of electron–hole recollision in these solids. This implies that gas-phase methods such as polarization gating for attosecond pulse generation and orbital tomography could be realized in solids.