Is there any daily limit for number push notifications (GCM) can be sent to a particular android application user?
After Google replaced C2DM with GCM, they took off all limits.
Prior to GCM (when it was still called C2DM): https://developers.google.com/android/c2dm/#limitations)
The only limits you run into the GCM documentation is this: http://developer.android.com/google/gcm/adv.html#lifetime
Quote from the above link:
Note: There is a limit on how many messages can be stored without collapsing. That limit is currently 100. If the limit is reached, all stored messages are discarded. Then when the device is back online, it receives a special message indicating that the limit was reached. The application can then handle the situation properly, typically by requesting a full sync.
I am not sure if this is relevant to your question though.