  We investigate circumstances under which one can generalize Horndeski's most
general scalar-tensor theory of gravity. Specifically we demonstrate that a
nonlinear combination of purely kinetic gravity terms can give rise to an
accelerating universe without the addition of extra propagating degrees of
freedom on cosmological backgrounds, and exhibit self tuning to bring a large
cosmological constant under control. This nonlinear approach leads to new
properties that may be instructive for exploring the behaviors of gravity.
