We have used the torsional oscillator technique to study supefluidity in 4He films adsorbed on the basal plane of graphite for coverages between zero and seven atomic layers. The development of superfluidity in this system is significantly different from that observed on disordered substrates such as Mylar and porous glasses. The most unusual phenomenon is a reentrant superfluid phase in the second layer of adsorbed helium. Superfluidity appears at a coverage just above one and one-half layers and then disappears before the completion of the second layer. We observe superfluidity again just above second layer completion, but its evolution with increasing coverage continues to be unusual. Plateaus in the superfluid transition temperature and superfluid signal occur over parts of the third and fourth layers. A periodic structure appears in the measured superfluid mass through the completion of the seventh layer.