We study the high-Reynolds-number behavior of a turbulent boundary layer in the low supersonic regime through very-large-scale direct numerical simulation (DNS). For the first time a Reynolds number is attained in DNS (Reτ = δ/δv ≈ 4000, where δ is the boundary layer thickness and δv is the viscous length scale) at which theoretical predictions and experiments suggest the occurrence of phenomena pertaining to the asymptotic Reynolds number regime. From comparison with previous DNS data at lower Reynolds number we find evidence of a continuing trend toward a stronger imprint of the outer-layer structures onto the near-wall region. This effect is clearly manifested both in flow visualizations, and in energy spectra. More than a decade of nearly-logarithmic variation is observed in the mean velocity profiles, with log-law constants k ≈ 0.394, C ≈ 4.84, and a trend similar to experiments. We find some supporting evidence for the debated existence of a k−1 region in the power spectrum of streamwise velocity fluctuations, which extends up to y+ ≈ 150, and of a k−5/3 spectral range in the outer layer.