Google Talk on iPhone (push notification)

I installed an app called Quicksilver for that, which was only a couple of bucks and works decently, and also downloaded the free version of Palringo. There's a number of other options from the App Store like Beejive, however. Just do a search for apps on the appstore with gtalk, jabber, or googletalk as the keywords.

These are all iPhone apps.

IM+ does the trick, on top of Google Talk it also does the trick for ICQ, MSN, Skype (chat), Yahoo, AIM, Twitter, Facebook, Jabber and Myspace...

Well, pretty much all the instant messaging apps for the iPhone allow you to remained signed into their servers after you've closed the app, pushing new messages via Apple's push services. However, I don't think any of them are going to let you stay signed in on their server for more than a day or so. I think the free version of IM+ lets you stay signed in for about 2 hours.