Android - "Android OS" is continuously downloading something in the background, how can I stop that?

I've found the culprit! I went to Settings > Call > Internet call settings and deleted the SIP account I had once created and forgot. The phone kept reaching the SIP server (probably to check for incoming calls).

I had to install a packet sniffer on the phone (tPacketCapture) to see where the requests where going and realize I had configured a SIP account.

Depending on what apps you have installed, apps like facebook, twitter, google+ and others will download data in the background to keep current when you open the app.

With ICS, you can disable background data usage for specific apps:

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this is visible in the system settings -> data usage. you should then see a list of apps that are using data. it will also show the highest usage app. Clicking on the app brings you to a details screen that allows you to disable data usage or background usage.

Have you looked at something like Onavo Count to monitor data usage?