Yes, there are a couple of ways that Aspire could be used to produce this, and yes, it's possible that a CNC machine could make it. If it were me, though, I would think about whether a CNC machine should make it. I suspect it would be easier and faster to make using more traditional techniques (table saw, router and lathe) than on a CNC, but I'd be open to discussion.
In Aspire, you could make a jig and cut the tapers as profile cuts on all four sides. The jig would need to support the angle once a side was cut away. You'd cut one side, flip the part 90 degrees, cut the next side and so on. The chamfers would be cut with a v-bit, using a fluting toolpath.
I suppose you could also create the bevel and chamfer with a 2-rail sweep and carving toolpath, for each side but it would take forever to cut it.
The finial could also be cut on a flat table by flipping the part as described above, either as a single sided part or as a 2-sided part (doesn't really matter as it's a regular shape all around). If you have a rotary attachment, it can also be designed and cut on that. I would not try to cut the long taper on a rotary attachment unless you have a steady rest, as the vibration is likely to damage the part.
D&S Artistic Woodworking http://www.dsartisticwood.com