  A novel, interesting class of scalar-tensor gravity theories is those with a
limit on the field motion, where the scalar field either goes to a constant
acceleration or stops accelerating and goes to a constant velocity. We combine
these with the ability to dynamically cancel a high energy cosmological
constant, e.g. through the well tempered or self tuning approaches. One can
successfully have a cosmic expansion history with a matter dominated epoch and
late time acceleration despite a large cosmological constant, although the late
time de Sitter limit may be unstable. Pole models, such as in a
Dirac-Born-Infeld action, are of particular interest for a cosmic speed limit.
