We study the dynamics of homogeneous, isotropic universes which are governed by the Eddington-inspired alternative theory of gravity which has a single extra parameter, κ. Previous results showing singularity-avoiding behavior for κ>0 are found to be upheld in the case of domination by a perfect fluid with equation of state parameter w>0. The range -130. Certain values of the parameters governing the potential lead to behavior in which the expansion rate of the universe changes sign several times before transitioning to regular general relativity-like behavior. © 2012 American Physical Society.