Tab Assassin (By Google)

Tab Assassin allows you to set a shelf life on your tabs. Don't use a tab for long enough, and Tab Assassin will silently and ruthlessly remove it. Not to fear, restoring that closed tab is just a couple of clicks away.

This is a solid, reliable extension that doesn't get in your way. Here are a few screenshots to show you how it looks:

Note: The development of this extension seemed abandoned and it has been resurrected by Divye Kapoor. If you're the original owner or Google's representative who'd like to take over the extension, please get in touch with me over email.