Rob, have you upgraded to Aperture 3.4.3? This is supposed to have fixed the serial number problem:
About Aperture 3.4.3
Addresses an issue that could cause a licensed copy of Aperture to prompt for a serial number with each launch
Be sure to enter the serial number, when you are logged in from an account with Administrator priveleges - otherwise can it not be stored.
Aperture is installed for all users on your mac, so the serial number is written to the system library, not the user library, and this requires read/write access for administrators. Do you enter your serial number using an administrator account?
When did this start? Have you migrated your system from from another mac? Then you still may have an older ProAppSystemID file or Aperture cannot write to the System Library.
To stop Aperture asking for the serial number, first check, if you still have an old file "ProAppSystemID" in your System Library.
Go to your MacintoshHD and open the folder "Library", then "Application Support", then "ProApps".
Remove the File "ProAppsSystemID" to the Desktop, if it exist and restart Aperture.
Reenter your serial number, hopefully for the last time.
Sometimes Aperture writes this file to a temporary directory, when it cannot write to the Library. If the above does not work, look at the temporary directory /tmp
Use the Finder's "Go" menu
Go > Got to folder and enter /tmp into the textfield. If you see the file there, move it to its proper location.
In this case you may want to repair the permissions on your system drive; you can do this using the "First Aid" tools in "Disk Utility" (located in Applications > Utilities)
Thanks for the quick respose. I am running 3.4.3, but I did not find "ProAppSystemID" in the system library. It was, rather, in /tmp with a date stamp corresponding to when I last re-entered the license last night, so I moved it (as root user) to the system library. So far so good, let's see if it stays fixed.