  We fabricate a twisted trilayer graphene device with consecutive twist angles
of 1.33 and 1.64 degrees, in which we electrostatically tune the electronic
states from each of the two co-existing moir\'e superlattices and the
interactions between them. When both moir\'e superlattices contribute equally
to electrical transport, we report a new type of inter-moir\'e Hofstadter
butterfly. Its Brown-Zak oscillation corresponds to one of the intermediate
quasicrystal length scales of the reconstructed moir\'e of moir\'e (MoM)
superlattice, shedding new light on emergent physics from competing atomic
orders.
