Good examples of Apps that do a great job of using push notifications to keep their customer bases active and engaged need to look no further than the Gaming Apps – especially those that you engage with socially. I have played/tested out the following games over the last few years; GinRummy, Words With Friends, Blackjack, Draw Something, Farmville, Trivia Crack and more.
What they all have in common it how they engage with their customer base and how those messages are trigged based on where you are in the game, if it is your turn, if something is triggered etc. It brings you back into the game to engage and to keep the App top of mind.
Download some games not just to play, but to see how they engage with you. Lots to learn from this category.