We characterize the amount of entanglement that is sufficient to play any XOR game near-optimally. We show that for any XOR game G and ε > 0 there is an ε -optimal strategy for G using [ε−1] ebits of entanglement, irrespective of the number of questions in the game. By considering the family of XOR games CHSH(n) introduced by Slofstra (Jour. Math. Phys. 2011), we show that this bound is nearly tight: for any ε > 0 there is an n = Θ(ε−1/5) such that Ω (ε −1/5) ebits are required for any strategy achieving bias that is at least a multiplicative factor (1- ε) from optimal in CHSH(n).