ZAPPA!! Now why didn't I list him?
Songs- Definately tell a story, most of which probably couldn't be listed here.
Composition- sometimes bizzare but genius!
And yes of course his "students" Satriani and Vai .
The List is heading towards favoring guitarist!!
Hagar- You listed Gene Krupa, so I was wondering about you playing drums, he is a major influance on many a great drummer including my favorite Peter Criss!