  We report the successful synthesis of single-crystalline cuprate
superconductors HgBa$_{2}$CaCu$_{2}$O$_{6+\delta}$ and
HgBa$_{2}$Ca$_{2}$Cu$_{3}$O$_{8+\delta}$. These compounds are well-known for
their high optimal superconducting critical temperatures of $T_\mathrm{c}$ =
128 K and 134 K at ambient pressure, respectively, and for their challenging
synthesis. Using a conventional quartz-tube encapsulation method and a
two-layer encapsulation method that utilizes custom-built high-pressure
furnaces, we are able to grow single crystals with linear dimensions up to
several millimeters parallel to the CuO$_2$ planes. Extended post-growth
anneals are shown to lead to sharp superconducting transitions, indicative of
high macroscopic homogeneity. X-ray diffraction and polarized Raman
spectroscopy are identified as viable methods to resolve the seemingly
unavoidable inter-growth of the two compounds. Our work helps to remove
obstacles toward the study of these model cuprate systems with experimental
probes that require sizable high-quality crystals.
