  Among various strong-curvature extensions to General Relativity,
Einstein-Dilaton-Gauss-Bonnet gravity stands out as the only nontrivial theory
containing quadratic curvature corrections while being free from the
Ostrogradsky instability to any order in the coupling parameter. We derive an
approximate stationary and axisymmetric black-hole solution of this
gravitational theory in closed form, which is quadratic in the black-hole spin
angular momentum and of seventh order in the coupling parameter of the theory.
This extends previous work that obtained the corrections to the metric only at
the leading order in the coupling parameter, and allows us to consider values
of the coupling parameter close to the maximum permitted by theoretical
constraints. We compute some geometrical properties of this solution, such as
the dilaton charge, the moment of inertia and the quadrupole moment, and its
geodesic structure, including the innermost-stable circular orbit and the
epicyclic frequencies for massive particles. The latter represent a valuable
tool to test General Relativity against strong-curvature corrections through
observations of the electromagnetic spectrum of accreting black holes.
