Very short arrays of continuous single-wall carbon nanotubes (SWNTs) are grown incrementally in steps as small as 25 nm using pulsed chemical vapor deposition (CVD). In-situ optical extinction measurements indicate that over 98% of the nanotubes reinitiate growth on successive gas pulses, and high-resolution transmission electron microscopy (HR-TEM) images show that the SWNTs do not exhibit segments, caps, or noticeable sidewall defects resulting from repeatedly stopping and restarting growth. Time-resolved laser reflectivity (3-ms temporal resolution) is used to record the nucleation and growth kinetics for each fast (0.2 s) gas pulse and to measure the height increase of the array in situ, providing a method to incrementally grow short nanotube arrays to precise heights. Derivatives of the optical reflectivity signal reveal distinct temporal signatures for both nucleation and growth kinetics, with their amplitude ratio on the first gas pulse serving as a good predictor for the evolution of the growth of the nanotube ensemble into a coordinated array. Incremental growth by pulsed CVD is interpreted in the context of autocatalytic kinetic models as a special processing window in which a sufficiently high flux of feedstock gas drives the nucleation and rapid growth phases of a catalyst nanoparticle ensemble to occur within the temporal period of the gas pulse, but without inducing growth termination.