  We investigate the electronic structure of a twisted multilayer graphene
system forming a moir\'e pattern. We consider small twist angles separating the
graphene sheets and develop a low-energy theory to describe the coupling of
Dirac Bloch states close to the K point in each individual plane. Extending
beyond the bilayer case, we show that, when the ratio of the consecutive twist
angles is rational, a periodicity emerges in quasimomentum space with moir\'e
Bloch bands even when the system does not exhibit a crystalline lattice
structure in real space. For a trilayer geometry, we find flat bands in the
spectrum at certain rotation angles. Performing a symmetry analysis of the band
model for the trilayer, we prove that the system is a perfect metal in the
sense that it is gapless at all energies. This striking result originates from
the three Dirac cones which can only gap in pairs and produce bands with an
infinite connectivity. The full gapless property is protected by an emergent
particle-hole symmetry valid at sufficiently small angles.
