We demonstrate the existence of two stable contact angles for the gallium droplet on top of self-assisted GaAs nanowires grown by MBE on patterned silicon substrates. Contact angle around 130 degrees fosters a continuous increase in the nanowire radius, while 90 degrees allows for the nanowire thinning, followed by the stable growth of ultra-thin tops. We develop a model that explains the observed morphological evolution under the two different scenarios.