I don't know what Google wants to be when it grows up, but I'll tell you one thing it shouldn't waste its time on along the way: Google IM. Sure, they did a great job with GMail, but e-mail is different. There are no network effects with e-mail. It doesn't matter what e-mail you use...your friends can always reach you. Instant messager is different. Trillian aside, you need to all be on the same system to instant message your friends. It remails the stickiest app on the web... and probably the only thing keeping a lot of people at AOL. I'll always use AOL IM because that's where all my friends are. I might use Yahoo, too, but only with Trillian.
Now, there have been rumors that Google will develop its own IM client, but I doubt this will take off. Perhaps they will offer Trillian like functionality--interoperability with other clients like MSN and Yahoo, but I doubt the other players will give up IM share that easily. While AOL and others seems to have stopped trying to block universal clients, they might start being defensive again if someone like a Google tried it... which is why I'm not advocating that Google just byuy Trillian.
But the #1 IM client in the US will be up for sale soon. Its only a matter of time before Time Warner waives the white flag and gives up on AOL. AOL would give Google a sticky IM client, and lots of content properties to run Google ads on... including all of those people still using AOL mail. Plus, given Google's stock price, I think now would be a good time to start using that capital for a big splash.
I think Google is probably the right company to take a shot at doing AOL the right way. All of their apps, be it AdSense, Maps, Gmail, have had a lot of time spent on them to make them as user friendly as possible--so from a user experience, I think the intentions are very similar. Remember, AOL is what introduced a lot of us to the web in the first place--we just grew out of it because of broadband.
Any thoughts on who else makes a good mate for the little yellow guy?