When an espionage operation occurs onto your nation
When it's your DC, or just before
When a trade offer is completed
Tells you how you've been attacked in war (Nuked, Missiles etc)
Resources received from alliance Bank
Treasures
War declarations
Embargoes
PW News
Messages
In-Game notifications, e.g baseball
If you can re-buy military
Reminders to tell you to log in (for the 500k$)
Alliance acceptance/denial
Vacation mode completion
Going out of beige
Project timer/city timer
Blockade broken
New treaties
Additionally users should be able to choose what notifications they get