  I discuss the relation between arbitrarily high-order theories of gravity and
scalar-tensor gravity at the level of the field equations and the action. I
show that $(2n+4)$-order gravity is dynamically equivalent to Brans-Dicke
gravity with an interaction potential for the Brans-Dicke field and $n$ further
scalar fields. This scalar-tensor action is then conformally equivalent to the
Einstein-Hilbert action with $n+1$ scalar fields. This clarifies the nature and
extent of the conformal equivalence between extended gravity theories and
general relativity with many scalar fields.
