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Interstellar dust extinction curves provide valuable information about dust properties, including the composition and size of the dust grains, and are essential to correct observations for the effects of interstellar dust. In this work, we measure a representative sample of near-infrared (NIR; 0.8-5.5 $μ$m) spectroscopic extinction curves for the first time, enabling us to investigate the extinction at wavelengths where it is usually only measured in broad photometric bands. We use IRTF/SpeX spectra of a sample of reddened and comparison stars to measure 15 extinction curves with the pair method. Our sample spans A(V) values from 0.78 to 5.65 and R(V) values from 2.43 to 5.33. We confirm that the NIR extinction curves are well fit by a power law, with indices and amplitudes differing from sight line to sight line. Our average diffuse NIR extinction curve can be represented by a single power law with index $α= 1.7$, but because of the sight line-to-sight line variations, the shape of any average curve will depend on the parental sample. We find that most of the variation in our sample can be linked to the ratio of total-to-selective extinction R(V), a rough measurement of the average dust grain size. Two sight lines in our sample clearly show the ice extinction feature at 3 $μ$m, which can be fitted by a modified Drude profile. We find tentative ice detections with slightly over 3$σ$ significance in two other sight lines. In our average diffuse extinction curve, we measure a 3$σ$ upper limit of A(ice)/A(V) = 0.0021 for this ice feature.