We present high-resolution spectroscopic observations of the quiet-Sun center-to-limb variations (CLV) of the He I triplet at 10 830 Å and the nearby Si I 10 827 Å line, observed with GREGOR Infrared Spectrograph (GRIS) and the improved High-resolution Fast Imager (HiFI+). The observations cover the interval μ = [0.1, 1.0], where μ is the cosine of the heliocentric angle. At each μ position, the spectra were spatially averaged over 0.02 μ and the resulting CLVs were given both as these averaged data points and as smooth polynomial curves fitted across each wavelength point. The He I spectra were inverted using the HAnle and ZEeman Light (HAZEL) code, showing an increase in optical depth towards the limb and a reversed convective blueshift for the red component, while the blue component was entirely absent. In addition, we found a strong increase in the steepness of the He I CLV compared to that of the nearby continuum. The Si I showed a behavior more typical of photospheric lines, namely shallower CLV, a reduction in width and depth, and a more typical convective blueshift.