They don't eject after I have logged in, or when I shutdown, only right before the login screen is displayed.
I do have a kensington wireless USB trackball and I read that same thing elsewhere. I removed the USB receiver and disconnected everything but the power cable, and it was still ejecting.
Last night I tried resetting the PRAM, and today I tried your suggestion about the other discs.
This Macbook Pro did not do this prior to the upgrade. It had Lion installed with it, and upgraded to Mountain Lion, no issues. It only started after the installation of Mavericks.
I have a bootcamp partition running Windows 7, and if I boot into it with the Game DVD in, it does not eject the DVD, and can read it just fine. Once I reboot into Mavericks, it ejects prior to login.