  We investigate the topological properties of Floquet-engineered twisted
bilayer graphene above the magic angle driven by circularly polarized laser
pulses. Employing a full Moir\'e-unit-cell tight-binding Hamiltonian based on
first-principles electronic structure we show that the band topology in the
bilayer, at twisting angles above 1.05$^\circ$, essentially corresponds to the
one of single-layer graphene. However, the ability to open topologically
trivial gaps in this system by a bias voltage between the layers enables the
full topological phase diagram to be explored, which is not possible in
single-layer graphene. Circularly polarized light induces a transition to a
topologically nontrivial Floquet band structure with the Berry curvature of a
Chern insulator. Importantly, the twisting allows for tuning electronic energy
scales, which implies that the electronic bandwidth can be tailored to match
realistic driving frequencies in the ultraviolet or mid-infrared photon-energy
regimes. This implies that Moir\'e superlattices are an ideal playground for
combining twistronics, Floquet engineering, and strongly interacting regimes
out of thermal equilibrium.
