Direct numerical simulations of turbulent pipe flow have been carried out up to friction Reynolds number Re_τ ≈ 12, 000. At those conditions the mean velocity distribution shows a distinct logarithmic layer, and a secondary peak of the streamwise velocity variance arises. The instantaneous flow visualizations and the spectral maps support decreasing influence of the superstructures on the near-wall flow. Correspondingly, the spectral imprint of the attached eddies exhibits a decline with the wavelength, possibly on account of viscous effects. Integration of the spectral densities leads to the conclusion that the streamwise velocity variance should stay finite in the infinite-Reynolds-number limit.