If you set up your email address to be copied into all system emails, you will get a copy of every registration acknowledgement. This can be done in System email settings on the Setup tab. Change the Admin email and turn on 'BCC admin' or 'BCC other' to receive a copy of every email sent by the system. Individual emails can be turned off above these settings.
Note that you will be copied into every system email and should also set up rules or filters on your email inbox to separate these from regular messages.