  Twisted bilayer graphene (tBLG) forms a quasicrystal whose structural and
electronic properties depend on the angle of rotation between its layers. Here
we present a scanning tunneling microscopy study of gate-tunable tBLG devices
supported by atomically-smooth and chemically inert hexagonal boron nitride
(BN). The high quality of these tBLG devices allows identification of
coexisting moir\'e patterns and moir\'e super-superlattices produced by
graphene-graphene and graphene-BN interlayer interactions. Furthermore, we
examine additional tBLG spectroscopic features in the local density of states
beyond the first van Hove singularity. Our experimental data is explained by a
theory of moir\'e bands that incorporates ab initio calculations and confirms
the strongly non-perturbative character of tBLG interlayer coupling in the
small twist-angle regime.
