The rarest fish currently in my possession Symphurus plagiusa, a small flatfish. The only time I have seen this species for sale was a listing on this site from nearly twenty years ago. I only have one at the moment. I have observed and caught these fish in their native environment several times in the past, though have not attempted to keep them yet. Mine is incredibly small, and I worry about its safety in my tank at times. However, they tend to appear in large quantities in late May into summertime, so I have little doubt that I will be able to obtain more.
Other bizarre native fish I have kept include Strongylura marina, Chasmodes bosquianus, Hypsoblennius hentz, and Prionotus carolinus.