Vai is just as big of an influence on me as Gilmour is, in some ways even more, especially with the more esoteric things Vai's been putting forth in the last few years. Love the way the guy just constantly wants to push himself, despite already making his mark.
There's so many little inflections I snagged from him, not so much technique stuff, but phrasing and noises, whammy abuse, song/solo structure. While the "best guitarist in the world' thing is silly, I can't say that his name doesn't immediately flash in my head when the topic is brought up.