  We report experimental results for heat-transport measurements by turbulent
Rayleigh-B\'enard convection in a cylindrical sample of aspect ratio $\Gamma
\equiv D/L = 1.00$ ($D = 1.12$ m is the diameter and $L = 1.12$ m the height).
They are for the Rayleigh-number range $4\times10^{11} \alt \Ra \alt
2\times10^{14}$ and for Prandtl numbers \Pra\ between 0.79 and 0.86.
  For $\Ra < \Ra^*_1 \simeq 2\times 10^{13}$ we find $\Nu = N_0
\Ra^{\gamma_{eff}}$ with $\gamma_{eff} = 0.321 \pm 0.002$ and $N_0 = 0.0776$,
consistent with classical turbulent Rayleigh-B\'enard convection in a system
with laminar boundary layers below the top and above the bottom plate and with
the prediction of Grossmann and Lohse.
  For $\Ra > \Ra_1^*$ the data rise above the classical-state power-law and
show greater scatter. In analogy to similar behavior observed for $\Gamma =
0.50$, we interpret this observation as the onset of the transition to the
ultimate state. Within our resolution this onset occurs at nearly the same
value of $\Ra_1^*$ as it does for $\Gamma = 0.50$. This differs from an earlier
estimate by Roche {\it et al.} which yielded a transition at $\Ra_U \simeq
1.3\times 10^{11} \Gamma^{-2.5\pm 0.5}$. A $\Gamma$-independent $\Ra^*_1$ would
suggest that the boundary-layer shear transition is induced by fluctuations on
a scale less than the sample dimensions rather than by a global
$\Gamma$-dependent flow mode. Within the resolution of the measurements the
heat transport above $\Ra_1^*$ is equal for the two $\Gamma$ values, suggesting
a universal aspect of the ultimate-state transition and properties. The
enhanced scatter of \Nu\ in the transition region, which exceeds the
experimental resolution, indicates an intrinsic irreproducibility of the state
of the system.
