A small portion of users on Xbox 360 have reported that they are stuck in a loop where they are constantly prompted to re-download the Xbox Live Compatibility Pack. To resolve this, please:
1) Turn off the Xbox 360.
2) Turn the Xbox 360 back on.
3) Press the Guide button on your controller, go to Settings, and select System Settings.
4) Select Storage or Memory.
5) Select the storage device containing the Compatibility Pack.
6) Select Games and Apps, then Borderlands 2.
7) Highlight each downloaded Compatibility Pack, press A and then select Delete and hit A again.
8) Boot Borderlands 2 while connected to Xbox Live.
9) When prompted, re-download the Compatability Pack.
10) Additionally, the title update may be corrupted. Clear the system cache of your Xbox 360. This will remove all downloaded game updates, and allow you to re-download any game updates that may be corrupted. You can find instructions here: http://support.xbox.com/en-US/xbox-360/settings-and-initial-setup/clear-system-cache
If you still experience the issue, please redownload the Compatibility Pack through your account history. You can find your Account History by navigating to the Settings Tab, selecting Account and Download History.
If the above steps do not resolve your problem, please submit a ticket for further assistance.