Re-rendering my scene with a Poser box that I put both displacement and transparency on just for the fun of it. (used a weave procedural and hooked that up to both the displacement and transparency channels) I also changed the rayBias on the mirror from 0.0125 to 0.1250 and it looks like it did the trick. Both the displacement and the transparency renders in the mirror now. I did change the raytrace bounces to 5 to make sure the transparency would render properly (the render setting says "4 for reflection through transparency" and I then used one extra just in case :))

So the verdict: PP2014 CAN do reflections of reflections. And it CAN do reflections of displacement. IF the RayBias is changed from its default setting of 0.0125 to something bigger 

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You just can't put the words "Poserites" and "happy" in the same sentence - didn't you know that? LaurieA
  Using Poser since 2002. Currently at Version 11.1 - Win 10.