  In cuprate superconductors, the highest superconducting transition
temperature $T_c$ is possessed by the HgBa$_2$Ca$_2$Cu$_3$O$_{8+\delta}$
(Hg-1223) system at ambient pressure, but the reason remains elusive. Here we
report the scanning tunneling microscope measurements on the Hg-1223 single
crystals with $T_c$ = 134 K. The observed superconducting gaps determined from
the tunneling spectra can be categorized into two groups: the smaller gap
$\Delta_1$ ranges from about 45 to 70 meV, while the larger gap $\Delta_2$ from
about 65 to 98 meV. The observed unprecedentedly large gap value gives a
straightforward explanation to the highest $T_c$ in the Hg-1223 system. The
largest gap observed here is comparable to the magnetic superexchange energy
and excludes any possibility of using phonon pictures to interpret the
superconductivity. Interestingly, an extremely strong particle-hole asymmetry
is observed in associating with a very robust coherence peak at the bias of the
larger gap in the hole branch of the Bogoliubov dispersion. We propose that the
observed asymmetry results from the interplay of a flat band (van Hove
singularity) in the electronic spectrum and the large superconducting gap in
the underdoped layer. This could be the main reason for the strong pairing, and
significant enhancement of the density of states in the hole branch of the
Bogoliubov band yielding strong phase coherence of Cooper pairs. A scenario
based on a trilayer model with an interlayer coupling can give a reasonable
explanation. Our results provide deep insight into understanding the mechanism
of superconductivity in cuprate superconductors.
