This was quite annoying, as there are many articles written about this, but only one solved it for myself.
The email@example.com email address is used by Exchange 2013 as part of the managed availability function. Health mailboxes are set up that ping the Exchange environment for health.
I found that I was getting undeliverable email bounces from my external mail gateway every 5 minutes (which is the timing of the Managed Availability pings). Which is annoying, as the queue will add up!
The one article that fixed the issue (thank you very much!)
Basically my Health mailboxes (which are created automatically) did not have a valid SMTP address of my FQDN of the forest root domain the Exchange environment was residing in. It had an external domain SMTP address, but the root domain was only a SIP address (See ADSI Edit under the proxyaddresses section of each Health Mailbox to verify).
Script in the article to add the valid root FQDN SMTP address to my health mailboxes resolved the issue.