Pattern formation in far-from-equilibrium systems is observed in several disciplines including biology, geophysics, and reaction-diffusion chemistry, comprising both living and nonliving systems. We aim to study such nonequilibrium dynamics on the laboratory scale with materials of simple composition. We present a novel system based on a 1-decanol droplet placed in a solution of alkaline decanoate. Previously, we showed the short time scale behavior of this system, which included chemotaxis and maze solving. Here we explore long time scale dynamics of the system (several hours) when open to the environment. We observe dramatic morphological changes in the droplet including long tentacular structures, and we analyze the morphology of these structures at both the macroscopic and microscopic scales across a large range of initial conditions. Such reproducible morphological changes in simple droplets open a path to the exploration of shape-based effects in larger-scale pattern-formation studies.