  We construct gravitational modifications that go beyond Horndeski, namely
theories with extended nonminimal derivative couplings, in which the
coefficient functions depend not only on the scalar field but also on its
kinetic energy. Such theories prove to be ghost-free in a cosmological
background. We investigate the early-time cosmology and show that a de Sitter
inflationary phase can be realized as a pure result of the novel gravitational
couplings. Additionally, we study the late-time evolution, where we obtain an
effective dark energy sector which arises from the scalar field and its
extended couplings to gravity. We extract various cosmological observables and
analyse their behavior at small redshifts for three choices of potentials,
namely, for the exponential, the power-law, and the Higgs potential. We show
that the Universe passes from deceleration to acceleration in the recent
cosmological past, while the effective dark-energy equation-of-state parameter
tends to the cosmological-constant value at present. Finally, the effective
dark energy can be phantom-like, although the scalar field is canonical, which
is an advantage of the model.
