The absorption of multiple photons when there is no resonant intermediate state is a non-linear process well known for atomic vapours, dyes and semiconductors. The N-Photon Absorption (NPA) rate for donors in semiconductors scales from hydrogenic atoms in vacuum proportionally with dielectric constant and inversely with effective mass -factors which carry exponents 6N and 4N respectivelyimplying extremely large enhancements are possible. We observed 1PA, 2PA and 3PA in Si:P with a THz free electron laser and e.g. the 2PA coefficient for 1s-2s at 4.25THz is 100,000,000GM (=10 -42 cm 4 s), many orders of magnitude larger than available in other systems. Such high cross-sections allowed us to enter a regime where the NPA cross-section exceeds that of 1PA, i.e. when the intensity approaches the binding energy per Bohr radius squared divided by the uncertainty time (only 3.84MW/cm 2 in silicon), and will enable new kinds of THz quantum control experiments.Multi-photon transitions 1 are valuable for high frequency resolution metrology 2,3 and high spatial resolution microscopes 4 . Photonic devices relying on such transitions include autocorrelators 5,6 , modulators 7,8 and sources of correlated photons 9 . The excitation rate for NPA is proportional to the intensity raised to the power of N for a drive frequency w eg /N where !w eg is the energy separation between the ground (g) and excited (e) states, and N is the number of photons required to achieve a Rydberg transition 1 . The non-linearity in intensity makes the absorption lines sharper, and this effect is therefore attractive for precise measurement of atomic level energies 2,3 . The non-linear coefficients for a hydrogenic oscillator scale with ! "# /% "#&' (see Eqn (2) below) where the effective dipole moment R is a weighted geometric mean of the dipole moments for each step in the N-photon process. In the case of 2PA this scaling is ! ( /%, so a system with small transition energy is attractive, and a system with a large dipole moment is very attractive. The silicon donor impurity is a hydrogenic atom with the outer electron orbiting a compact, singly and positively charged ion core. The Coulomb force is greatly reduced by the dielectric screening in the semiconductor environment, as is the effective mass, and the combined effect is to lower the binding energy (Fig 1) and produce enlarged, mesoscale wave functions with an effective Bohr radius of 2 3.17nm. This results naturally in the desired high R. Here we investigate one-, two-, and three-photon transitions for impurities in silicon which provide giant values of ! "# /% "#&' , comparable or larger even than systems with a near resonant intermediate state 5,6,7,10,11 . Indeed in our experiment the signal to noise of the 2PA is comparable to that for 1PA. Although these experiments made use of the flexible tuning and high power of the free electron laser, ideal for THz non-linear optics experiments, the high coefficients we found open the possibility of high efficiency devices such as photon pair e...