  The most general class of 4D N=4 conformal supergravity actions depends on a
holomorphic function of the scalar fields that parametrize an SU(1,1)/U(1)
coset space. The bosonic sector of these actions was presented in a letter
[arXiv:1609.09083]. Here we provide the complete actions to all orders in the
fermion fields. They rely upon a new N=4 density formula, which permits a
direct but involved construction. This density formula also recovers the
on-shell action for vector multiplets coupled to conformal supergravity.
Applications of these results in the context of Poincar\'e supergravity are
briefly discussed.
