In "How to Email Busy People", startup founder Jason Freedman gives several good tips on getting what you want from busy and important people. The most crucial tip, as quoted today by designer Tina Roth Eisenberg, is to be explicit instead of coy.
If you want a meeting, ask for a meeting. Provide some time options and ask for a specified length. If you want an introduction, ask for an introduction. If you're looking for funding, tell him you're currently fundraising and ask to meet to show him your pitch. Don't be sly. Don't hint. Make the process ridiculously easy by just asking for what you want.
If you're emailing someone you haven't talked to before, and you're trying to "build up to" your big ask, don't. You're not being polite, you're just being scared, and you're making your recipient do the emotional work of coaxing your real question out of you.
If you're buried under unanswered email, and find yourself constantly starting your replies with 'Sorry for the delay,' do what BuzzFeed reporter and creative miscreant Katie Notopoulos did. Answer your emails right away, with just a couple of words. She calls it 'emailing like a CEO', the same phrase used in a 2001 New York Times piece about how high-tier executives tend to send terse, misspelled emails.
Busy people want short, direct emails. Not quickly written emails, but quickly read emails. Follow Freedman's six other tips, and check Lifehacker's advice on efficiently and respectfully asking for favours.