The uncatalyzed edge growth of carbon nanotubes was investigated by first-principles molecular dynamics simulations. At experimental temperatures the open end of single-walled nanotubes closed spontaneously into a graphitic dome, which may explain why these nanotubes do not grow in the absence of transition metal catalysts, On the other hand, chemical bonding between the edges of adjacent coaxial tubes ("lip-lip" interactions) trapped the end oi a double-walled nanotube in a metastable energy minimum, thus preventing dome closure. These calculations show that this end geometry exhibits a high degree of chemical activity and easily accommodates incoming carbon fragments, supporting a model of growth by chemisorption from the vapor phase.