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Tevelev degrees in Gromov-Witten theory are defined whenever there are virtually a finite number of genus $g$ maps of fixed complex structure in a given curve class $\beta$ through $n$ general points of a target variety $X$. These virtual Tevelev degrees often have much simpler structure than general Gromov-Witten invariants. We explore here the question of the enumerativity of such counts in the asymptotic range for large curve class $\beta$. A simple speculation is that for all Fano $X$, the virtual Tevelev degrees are enumerative for sufficiently large $\beta$. We prove the claim for all homogeneous varieties and all hypersurfaces of sufficiently low degree (compared to dimension). As an application, we prove a new result on the existence of very free curves of low degree on hypersurfaces in positive characteristic., Comment: final version, erroneous example removed, reference to forthcoming work of Beheshti-Lehmann-Riedl-Starr-Tanimoto added. To appear in Ann. Sc. Norm. Super. Pisa Cl. Sci