One of the central goals of ecology is to predict the distribution and abundance of organisms. Here, we show that, in ecosystems of high biodiversity, the outcome of multispecies competition can be fundamentally unpredictable. We consider a competition model widely applied in phytoplankton ecology and plant ecology in which multiple species compete for three resources. We show that this competition model may have several alternative outcomes, that the dynamics leading to these alternative outcomes may exhibit transient chaos, and that the basins of attraction of these alternative outcomes may have an intermingled fractal geometry. As a consequence of this fractal geometry, it is impossible to predict the winners of multispecies competition in advance.