We demonstrate that gravitational waves generated by efficient gauge preheating after axion inflation generically contribute significantly to the effective number of relativistic degrees of freedom $N_mathrm{eff}$. We show that, with existing Planck limits, gravitational waves from preheating already place the strongest constraints on the inflatons possible axial coupling to Abelian gauge fields. We demonstrate that gauge preheating can completely reheat the Universe regardless of the inflationary potential. Further, we quantify the variation of the efficiency of gravitational wave production from model to model and show that it is correlated with the tensor-to-scalar ratio. In particular, when combined with constraints on models whose tensor-to-scalar ratios would be detected by next-generation cosmic microwave background experiments, $rgtrsim 10^{-3}$, constraints from $N_mathrm{eff}$ will probe or rule out the entire coupling regime for which gauge preheating is efficient.
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