  We present new examples of topologically convex edge-ununfoldable polyhedra,
i.e., polyhedra that are combinatorially equivalent to convex polyhedra, yet
cannot be cut along their edges and unfolded into one planar piece without
overlap. One family of examples is acutely triangulated, i.e., every face is an
acute triangle. Another family of examples is stacked, i.e., the result of
face-to-face gluings of tetrahedra. Both families achieve another natural
property, which we call very ununfoldable: for every $k$, there is an example
such that every nonoverlapping multipiece edge unfolding has at least $k$
pieces.
