The nature of fractional quantum Hall (FQH) states is determined by the interplay between the Coulomb interaction and the symmetries of the system. The distinct combination of spin, valley, and orbital degeneracies in bilayer graphene is predicted to produce an unusual and tunable sequence of FQH states. Here, we present local electronic compressibility measurements of the FQH effect in the lowest Landau level of bilayer graphene. We observe incompressible FQH states at filling factors n = 2p + 2/3, with hints of additional states appearing at v = 2p + 3/5, where p = -2, -1, 0, and 1. This sequence breaks particle-hole symmetry and obeys a v -> v + 2 symmetry, which highlights the importance of the orbital degeneracy for many-body states in bilayer graphene.