An investigation of the evolution of strongly nonlinear, low frequency (~ion gyrofrequency), parallel propagating wave packets in a dispersive, collisionless, and low &bgr;(=8&pgr;P/B2=0.3) plasma is undertaken using a hybrid numerical code. These strongly nonlinear wave packets have a transverse magnetic field strength, or wave amplitude, which is of order or greater than the field strength along the direction of propagation, and their evolution can differ qualitatively from that of weakly nonlinear packets. Strongly nonlinear, Alfven wave packets steepen but not from behind the maximum of wave amplitude as occurs during wave collapse. The development of spreading fast wave (right helicity) and rarefaction regions competes strongly with steepening, and leads to a long time waveform which differs greatly from that for weak nonlinearity. Results are used to suggest that strongly nonlinear wave evolution occurs frequently in the Earth's foreshock. ¿ American Geophysical Union 1992