Most of us here already realize that one of the best ways of generalizing antiprisms to higher dimensions is to analyze the top/bottom polygons of the 3D antiprisms as duals, rather than rotated copies of each other. This allows convenient definition of higher-dimensional antiprisms such as the cube antiprism (x4o3o || o4o3x), and so forth. Since the dual of a polytope always exists (though it may not always be CRF), this construction will always yield a polytope, that moreover preserves the symmetries of the base polytope.

Today I started wondering, though: does this definition really allow unbounded extension of the concept of antiprisms to higher dimensions? Specifically, I considered the line of n-cube antiprisms. One thing that immediately occurred to me is that the circumradius of the n-cube increases without bound as n increases, but the circumradius of the n-cross remains constantly at 1/√2. So there must exist some value of n above which the n-cube antiprism can no longer be CRF-able, because it would not be possible for the lacing edges to be unit length. A quick test computation revealed that this threshold is as low as 5: the tesseract antiprism is the last CRF in the series! The circumradius of the 5-cube is too large for unit lacing edges with the 5-cross, and therefore the 6D 5-cube antiprism cannot be CRF.

This leaves not many options left for constructing antiprisms (of this type). It's already well-known that the n-simplex antiprism always exists, and is CRF -- in fact, it's regular, and is identical to the (n+1)-cross. Among the uniform polytopes, this seems to be the only one that continues indefinitely. In 5D there's also the 24-cell antiprism which should be CRF, probably scaliform? I'm almost certain the 120-cell/600-cell antiprism cannot be CRF, due to the large discrepancy in circumradii. I don't know of many 5D CRFs that have CRF duals, but it would seem there wouldn't be very many of them, so they wouldn't have corresponding higher-dimensional CRF antiprisms.

The 5D tesseract antiprism is an interesting polyteron. It consists of 8 cubical pyramids, 16+32=48 5-cells, and 24 square-pyramid pyramids (square||line), for a total of 80 facets. Initially I thought there'd be some interesting occurrences of lower-dimensional antiprisms in it, but it appears that I may have been mistaken.

Less symmetric antiprisms may go a bit further. I haven't studied them as much yet. In 4D there are the n-pyramid antiprisms, which are just antiprism bipyramids. But I don't know of any non-regular self-dual 4D CRFs that could serve as the basis for 5D CRF antiprisms. Generally it seems the higher the dimension, the harder it is to construct CRF antiprisms of this type.