I run a VPN service called DataBuster. It is a simple adblocker for iOS devices. It blocks requests at the DNS level, so it is network-wide and thus very efficient.
The connections are strictly anonymous and very secure (the code is here), but I anonymously log the DNS requests. It is a way of monitoring the usages of my servers.
When I tested the service on my device, I discovered that iOS communicated non-stop with Apple servers. I thought it was linked to the Push services, but I looked into the logs and here are the concerned domains that are iOS-related:
apple.com icloud.com akamaiedge.net akamadns.net apple-dns.net akamai.net akadns.net cdn-apple.com aaplimg.com
Here is a screenshot showing the unexpected number of requests your phone makes. Notice the activity between 3AM and 5AM (I swear I was sleeping haha).
I haven't tried on Android devices, but it should be interesting as well. Another experiment would be to block these domains and see the impact on your battery life ;)
Here is what it looks like with more devices. Observe the difference of requests between the two graphs.
Here is a breakdown of all concerned hosts in a light csv file.
Read more on the Hacker News discussion.