I had an instance where I needed to send multiple emails any time a new user registered as there were a few different processes that needed to happen to onboard a new user. I accomplished this by adding a custom options page and custom fields to hold the email addresses so they could be easily changed, then hooking to the user_register event and gathering those emails to also send out the new user's information. This functionality could easily be extended or developed as a plugin to handle other email notifications as well. You could even pull email addresses directly from your Users info based on their account permissions or roles depending on how you need this functionality to work.
If you don't want to write the PHP for it, you can easily create custom options pages with custom fields using plugin ACF with the Option Page addon. http://www.advancedcustomfields.com/
My first instinct was to just edit the PHP in /wp-includes/pluggable.php which is where the emails are actually fired, however I suggest against this as any upgrade to the WordPress core will overwrite your modifications.