Thermal Hall conductivity in the strongest cuprate superconductor : Estimate of the mean free path in the trilayer cuprate HgBa <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi/><mml:mn>2</mml:mn></mml:msub></mml:math> Ca <mml:math xmlns:mml="http://www.w3.

The thermal Hall conductivity of the trilayer cuprate HgBa$_2$Ca$_2$Cu$_3$O$_{8+\delta}$ (Hg1223) - the superconductor with the highest critical temperature $T_c$ at ambient pressure - was measured at temperatures down to 2 K for three dopings in the underdoped regime ($p$ = 0.09, 0.10, 0.11). By combining a previously introduced simple model and prior theoretical results, we derive a formula for the inverse mean free path, $1 / \ell$, which allows us to estimate the mean free path of $d$-wave quasiparticles in Hg1223 below $T_c$. We find that $1 / \ell$ grows as $T^3$, in agreement with the theoretical expectation for a clean $d$-wave superconductor. Measurements were also conducted on the single layer mercury-based cuprate HgBa$_2$CuO$_{6+\delta}$ (Hg1201), revealing that the mean free path in this compound is roughly half that of its three-layered counterpart at the same doping ($p$ = 0.10). This observation is be attributed to the protective role of the outer planes in Hg1223, which results in a more pristine inner plane. We also report data in an ultraclean crystal of YBa$_2$Cu$_3$O$_y$ (YBCO) with full oxygen content $p$ = 0.18, believed to be the cleanest of any cuprate, and find that $\ell$ is not longer than in Hg1223.