PROMPT (Panchromatic Robotic Optical Monitoring and Polarimetry Telescopes) observed the early-time optical afterglow of GRB 060607A and obtained a densely sampled multiwavelength light curve that begins only tens of seconds after the GRB. Located at Cerro Tololo Inter-American Observatory in Chile, PROMPT is designed to observe the afterglows of gamma-ray bursts using multiple automated 0.4-m telescopes that image simultaneously in many filters when the afterglow is bright and may be highly variable. The data span the interval from 44 seconds after the GRB trigger to 3.3 hours in the Bgri filters. We observe an initial peak in the light curve at approximately three minutes, followed by rebrightenings peaking around 40 minutes and again at 66 minutes. Although our data overlap with the early Swift gamma-ray and x-ray light curves, we do not see a correlation between the optical and high-energy flares. We do not find evidence for spectral evolution throughout the observations. We model the variations in the light curves and find that the most likely cause of the rebrightening episodes is a refreshment of the forward shock preceded by a rapidly fading reverse shock component, although other explanations are plausible.