We compute the gravitational wave energy Erad radiated in head-on collisions of equal-mass, nonspinning black holes in up to (D =8 )-dimensional asymptotically flat spacetimes for boost velocities v up to about 90% of the speed of light. We identify two main regimes: weak radiation at velocities up to about 40% of the speed of light, and exponential growth of Erad with v at larger velocities. Extrapolation to the speed of light predicts a limit of 12.9% (10.1, 7.7, 5.5, 4.5)% of the total mass that is lost in gravitational waves in D =4 (5, 6, 7, 8) spacetime dimensions. In agreement with perturbative calculations, we observe that the radiation is minimal for small but finite velocities, rather than for collisions starting from rest. Our computations support the identification of regimes with super-Planckian curvature outside the black-hole horizons reported by Okawa, Nakao, and Shibata [Phys. Rev. D 83, 121501(R) (2011), 10.1103/PhysRevD.83.121501].