With iPhone or iPod Touch using GMail, all you need is OS 3.1 and an app called GPush which is $.99 off the App Store. Yes, it will push and give you a home screen notify, plus some blinking audible sound.
Boxcar, which is another paid but cheap app, you can add push notifications on Twitter and Facebook. Add .99 cents for every email account. So if you have five email accounts, you're paying five dollars for a one time fee.
There are other push apps in the app store, such as Notification.
I don't have an iPhone but I got an iPod Touch. When in range of Wifi, I can often hear the device go blink blink with every new email, Twitter or Facebook update, as well as various breaking news updates.
On the iPhone and iPod Touch, other apps have their own system of notifications.
CNN Mobile News app
MSNBC mobile news app
If there is breaking news for example, the app displays a message on the screen with some blink blink audible notification.
What the iPhone OS lacks is a unified message box. The push notifications disappear from the screen when other app sends the next notification. As a result there is no activity log of incoming notifications to review. The home screen only displays the most recent notification.
All these notifications do not need some Exchange sync.
I am @guamguy at Twitter.