We demonstrate that unidirectionally aligned, free-standing multi-walled carbon nanotubes ( CNTs) exhibit highly anisotropic linear and nonlinear terahertz ( THz) responses. For the polarization parallel to the CNT axis, strong THz pulses induce nonlinear absorption in the quasi-one-dimensional conducting media, while no nonlinear effect is observed in the perpendicular polarization configuration. Time-resolved measurements of transmitted THz pulses and a theoretical analysis of the data reveal that intense THz fields enhance permittivity in carbon nanotubes by generating charge carriers.