An abnormal fear of thunder and lightning, quite common in dogs.  Also known as astraphobia, keraunophobia, or tonitrophobia (from Wikipedia).  None of those terms, nor brontophobia, is considered an acceptable word by this spell checker, by the way.

Robbie points out that someone scared of phobias would be phobiaphobic, also, undoubtedly, not a word.

Bonus question: What is a cailihpphiliac?