How do I create a link that points users to the edit profile page? I would like to create a link that leads each user to their edit profile page.

What should I use as URL? I get access denied on the paths I tried, including http://example.com/user/[uid]/edit.
You might want to try using the Menu Token module, so you could add a user id token in the path.

Otherwise you'd need to create a page callback in a custom module using hook_menu(), say to 'user/edit', and in that callback, redirect the user to 'user/[uid]/edit'. That way you could add a menu link to 'user/edit' and it would work. See this comment for an example.