  Layers of two-dimensional materials stacked with a small twist-angle give
rise to beating periodic patterns on a scale much larger than the original
lattice, referred to as a moir\'e superlattice. When the stacking involves more
than two layers with independent twist angles between adjacent layers, it
generates moir\'e of moir\'e superlattices, with multiple length scales that
control the system's behavior. Here we demonstrate these effects of a
high-order moir\'e superlattice in twisted trilayer graphene with two
consecutive small twist angles. We report correlated insulating states near the
half filling of the moir\'e of moir\'e superlattice at an extremely low carrier
density (~1010 cm-2), near which we also report a zero-resistance transport
behavior typically expected in a 2D superconductor. Moreover, the temperature
dependence of the measured resistances at full-occupancy (v = -4 and v = 4)
states are semi-metallic, distinct from the insulating behavior of twisted
bilayer systems, providing the first demonstration of emergent correlated
transport behaviors from continuous, non-isolated higher-order moir\'e flat
bands. Our findings shed new insights into the microscopic mechanisms of
moir\'e correlated states and provide the impetus for future studies on this
material platform, such as the demonstration of phase coherence and
Meissner-like effect.
