Clusters of galaxies are excellent locations to probe the distribution of baryons and dark matter (DM) over a wide range of scales. We study a sample of seven massive (M 200 = 0.4-2 × 1015 M ⊙), relaxed galaxy clusters with centrally located brightest cluster galaxies (BCGs) at z = 0.2-0.3. Using the observational tools of strong and weak gravitational lensing, combined with resolved stellar kinematics within the BCG, we measure the total radial density profile, comprising both dark and baryonic matter, over scales of ~= 3-3000 kpc. We present Keck spectroscopy yielding seven new spectroscopic redshifts of multiply imaged sources and extended stellar velocity dispersion profiles of the BCGs. Lensing-derived mass profiles typically agree with independent X-ray estimates within ~= 15%, suggesting that departures from hydrostatic equilibrium are small and that the clusters in our sample (except A383) are not strongly elongated or compressed along the line of sight. The inner logarithmic slope γtot of the total density profile measured over r/r 200 = 0.003-0.03, where ρ_{tot} ∝ r^{-γ_{tot}}, is found to be nearly universal, with a mean langγtotrang = 1.16 ± 0.05(random)+0.05 -0.07 (systematic) and an intrinsic scatter σγ < 0.13 (68% confidence). This is further supported by the very homogeneous shape of the observed velocity dispersion profiles, which are mutually consistent after a simple scaling. Remarkably, this slope agrees closely with high-resolution numerical simulations that contain only DM, despite the significant contribution of stellar mass on the scales we probe. The Navarro-Frenk-White profile characteristic of collisionless cold DM is a better description of the total mass density at radii >~ 5-10 kpc than that of DM alone. Hydrodynamical simulations that include baryons, cooling, and feedback currently provide a poorer match. We discuss the significance of our findings for understanding the physical processes governing the assembly of BCGs and cluster cores, particularly the influence of baryons on the inner DM halo.