In this paper we formulate and analyze an N-player stochastic game of the classical fuel follower problem and its mean field game (MFG) counterpart. For the N-player game, we obtain the Nash equilibrium (NE) explicitly by deriving and analyzing a system of Hamilton-Jacobi-Bellman equations and by establishing the existence of a unique strong solution to the associated Skorokhod problem on an unbounded polyhedron with an oblique reflection. For the MFG, we derive a bang-bang type NE under some mild technical conditions and by the viscosity solution approach. We also show that this solution is an epsilon-NE to the N-player game, with epsilon = O(1/root N). The N-player game and the MFG differ in that the NE for the former is state dependent while the NE for the latter is a threshold-type bang-bang policy where the threshold is state independent. Our analysis shows that the NE for a stationary MFG may not be the NE for the corresponding MFG.