First off don't worry! This is not usually a sign there is something wrong with your console or memory device.
Having to repeatedly download updates every time you play any game is usually a symptom that your cache is full. Because there is only a finite amount of space made available for updates, there is simply no room left to save these downloads. Usually this occurs when you have played a lot of different titles over time and have never cleared the cache before.
The first step in fixing this issue is to clear your cache.
(Just to let you know beforehand, you will not lose your profile, game saves, multiplayer stats or any game-related DLC you have on your memory device. Sometimes gamer pictures can be stored on the cache, so you may need to re download those from your 'download history' after clearing).
Now to clear your cache:
Sign in on your console and on the 'Home' hub of the dashboard and scroll to the far right, then select the 'Settings' hub.
Then select the 'System' tile in the upper left.
From here scroll down x2 and select 'memory.'
Highlight the HDD and press 'Y', then select 'clear system cache.'
(A message will pop up saying 'This will perform maintainance on your xbox 360 storage device, do you want to continue?'
Now that you've cleared the cache, pop a game in your console and install the update when prompted. You will need to perform this for every title you play after clearing, but it should only be the once per title now (unless future
updates are released).
If you're still having to constantly re-download updates then please make a post in the forums, but this has always resolved the issue when I have experienced it myself.Space.