The latest giant in the world of free or cheap web-based emails is Gmail. Supported by a company billions of Google dollars, you will think they can buy a simple mechanism to look for any Gmail address in their database.
Unfortunately, there are no search tools like that, so that individuals who want to find the long Gmail address are left to try to make an inventive way to track their friends. Even though there is no method that is guaranteed to work, they might be your best bet.
Assuming you know that person, you can go to various websites that you know using it or you think it's possible they will use. It could be a site that is related to one of their main interests or can be a site to play games or even social networks.
These sites often have members search features where you can search for someone's name or screen name in the hope of finding an email address. It is still a shot in the dark, but some of these sites, like Facebook, print the user's email address right on their yard, so assuming you can find it, it must be there.
If the person you are looking for any clubs or organizations online, you can visit the webpage for the group and search for members. Often, each member will have its own page with registered basic contact information so that it finds an email address this way must be straight forward.