A variety of nonequilibrium growth structures is observed in fatty acid ethyl ester monolayers. In these monolayers, two different condensed phases exist : one with tilted chains and the other with erected chains. The formation of the domains of these phases from the fluid phase gives rise to completely different shapes under non-equilibrium. In the phase with tilted chains, a dendritic growth regime exists in ethyl stearate monolayers for intermediate compression rates. The arms of the dendrite have a tip-oscillating morphology. At extremely high compression rates the tips are no longer stable and the growth does not follow crystallographic directions. For the shorter chain (ethyl palmitate) the dendritic growth regime is not observed in the phase with tilted chains at low temperatures. Here the domain wall only becomes unstable at extremely high compression rates and the morphology is then fractal-like. However, at temperatures T > 20 degrees C a dendritic growth regime occurs. The nonequilibrium structures in both ethyl stearate and ethyl palmitate monolayers are transformed into a circular equilibrium shape for the condensed phase with tilted chains during a few minutes. In ethyl palmitate monolayers, a phase with erected chains forms during the first-order phase transition at temperatures T > 26 degrees C. In this phase the nonequilibrium structures are quite stable and transform very slowly into a compact shape. Even at extremely low compression rates branched structures occur. An increase of the compression rate causes a continuous change from a more dendritic shape to one which is quite similar to the dense-branching morphology.