Random Product Idea #1: Inbox100. It’s an email inbox where it has a hard upper limit of 100 messages. Once you hit 100 messages, until you archive or delete some current messages, no new emails come in. This forces you to deal with emails rather than allowing them to pile up. And the instant you deal with emails, you get the positive reinforcement of seeing new emails appear. You would probably still need to allow the user to search emails, even the ones not being shown, so that the user could look for super-important, super-urgent emails.
This leads me to Random Product Idea #2: InboxNow. It’s an email inbox where it only shows messages that are less than 12 hours old (because presumably you have to sleep). This forces you to deal with emails rapidly, or they disappear.