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We report the temperature dependence of the upper critical fields $\mu_0H_{c2}(T)$ of the high temperature superconductor H$_3$S under applied pressures of 155 and 160 GPa through the electrical resistance transition observed under DC and pulsed magnetic fields up to 65 T, a record high combination of fields and pressures. We find that $H_{c2}(T)$ generally follows the Werthamer, Helfand and Hohenberg (WHH) formalism at low fields, albeit with noticeable deviations upon approaching our experimental limit of $\mu_0H = 65$ T. In fact, $H_{c2}(T)$ displays a remarkably linear dependence on temperature over an extended temperature range also found in multigap or in strongly-coupled superconductors. The best fit of $H_{c2}(T)$ to the WHH formula yields a negligible value for the Maki parameter $\alpha$ and for spin-orbit scattering constant $\lambda_{\text{SO}}$. However, its behavior is relatively well-described by a model based on strong coupling superconductivity with a coupling constant $\lambda \sim 2$. Therefore, we conclude that H$_3$S behaves as a strong-coupled orbital-limited superconductor over the entire range of temperatures and fields used for our measurements.