The absorption of multiple photons when there is no resonant intermediate state is a well-known nonlinear process in atomic vapours, dyes and semiconductors. The N-photon absorption (NPA) rate for donors in semiconductors scales proportionally from hydrogenic atoms in vacuum with the dielectric constant and inversely with the effective mass, factors that carry exponents 6N and 4N, respectively, suggesting that extremely large enhancements are possible. We observed 1PA, 2PA and 3PA in Si: P with a terahertz free-electron laser. The 2PA coefficient for 1s-2s at 4.25 THz was 400,000,000 GM (= 4 x 10(-42) cm(4) s), many orders of magnitude larger than is available in other systems. Such high cross-sections allow us to enter a regime where the NPA cross-section exceeds that of 1 PA-that is, when the intensity approaches the binding energy per Bohr radius squared divided by the uncertainty time (only 3.84 MW cm(-2) in silicon)-and will enable new kinds of terahertz quantum control.