The answer depends on the kind of battery chemistry and how literally you mean "rare earth". If you take some slack on the definition and just mean "metal stuff in limited supply", then many battery chemistries have limited supplies.

There are, however, some chemistries with really nice supply chains. The Iron Redox Flow Battery (IRFB) really only needs iron and iron chloride as reactants. Those batteries are being commercialized, but they aren't common (yet?).