My website is currently Google PR5. How many directories would I need to submit to in order to go to a PR6 or 7?
There is no specific number, and alot of it has to do with the content of your website. You should base the number of your directory submissions on quality not quantity.
My experience is that you can get to a PR3 or 4 with web directories, but you need other types of links to break the PR5 barrier. So if you're looking for PR you should spend your time trying to attract links from other high PR pages.
I wouldn't recommend spending time to go from PR5 to PR6-PR7 by using directories. Breaking that barrier is a little harder and will take more stronger, practical links.