  Some Stein manifolds (with a volume form) have a large group of
(volume-preserving) automorphisms: this is formalized by the (volume) density
property, which has remarkable consequences. Until now all known manifolds with
the volume density property are algebraic, and the tools used to establish this
property are algebraic in nature. In this note we adapt a known criterion to
the holomorphic case, and give the first known examples of non-algebraic
manifolds with the volume density property: they arise as suspensions or
pseudo-affine modifications over Stein manifolds satisfying some technical
properties. As an application we show that there are such manifolds that are
potential counterexamples to the Zariski Cancellation Problem, a variant of the
Toth-Varolin conjecture, and the problem of linearization of C*-actions on C^3.
