We present high-precision measurements of the Nusselt number $\cal{N}$ as a function of the Rayleigh number $R$ for cylindrical samples of water (Prandtl number $\sigma \,{=}\, 4.38$) with diameters $D \,{=}\, 49.7, 24.8,$ and 9.2cm, all with aspect ratio $\Gamma \,{\equiv}\, D/L \,{\simeq}\,1$ ($L$ is the sample height). In addition, we present data for $D \,{=}\, 49.7$ and $\Gamma \,{=}\, 1.5, 2, 3,$ and 6. For each sample the data cover a range of a little over a decade of $R$. For $\Gamma \,{\simeq}\,1$ they jointly span the range $10^7 \lesssim R \lesssim 10^{11}$. Where needed, the data were corrected for the influence of the finite conductivity of the top and bottom plates and of the sidewalls on the heat transport in the fluid to obtain estimates of $\cal{N}_{\infty}$ for plates with infinite conductivity and sidewalls of zero conductivity. For $\Gamma \,{\simeq}\,1$ the effective exponent $\gamma_{\hbox{\scriptsize\it eff}}$ of ${\cal N}_{\infty} \,{=}\, N_0 R^{\gamma_{\hbox{\scriptsize\it eff}}}$ ranges from 0.28 near $R \,{=}\, 10^8$ to 0.333 near $R \,{\simeq}\,7\,{\times}\,10^{10}$. For $R \lesssim 10^{10}$ the results are consistent with the Grossmann–Lohse model. For larger $R$, where the data indicate that ${\cal N}_\infty(R) \,{\sim}\, R^{1/3}$, the theory has a smaller $\gamma_{\hbox{\scriptsize\it eff}}$ than $1/3$ and falls below the data. The data for $\Gamma \,{>}\, 1$ are only a few percent smaller than the $\Gamma \,{=}\, 1$ results.