  We present comprehensive theoretical studies on the lattice relaxation and
the electronic structures in general non-symemtric twisted trilayer graphenes.
By using an effective continuum model, we show that the relaxed lattice
structure forms a patchwork of moir\'e-of-moir\'e domains, where a moir\'e
pattern given by layer 1 and 2 and another pattern given by layer 2 and 3
become locally commensurate. The atomic configuration inside the domain
exhibits a distinct contrast between chiral and alternating stacks, which are
determined by the relative signs of the two twist angles. In the chiral case,
the electronic band calculation reveals a wide energy window ($>$ 50 meV) with
low density of states, featuring sparsely distributed highly one-dimensional
electron bands. These one-dimensional states exhibit a sharp localization at
the boundaries between super-moir\'e domains, and they are identified as a
topological boundary state between distinct Chern insulators. The alternating
trilayer exhibits a coexistence of the flat bands and a monolayer-like Dirac
cone, and it is attributed to the formation of moir\'e-of-moir\'e domains
equivalent to the mirror-symmetric twisted trilayer graphene.
