Node.js Web Application examples/tutorials
I would suggest you check out the various tutorials that are coming out lately. My current fav is:
http://nodetuts.com/
Hope this helps.
Update
Dav Glass from Yahoo has given a talk at YuiConf2010 in November which is now available in Video from.
He shows to great extend how one can use YUI3 to render out widgets on the server side an make them work with GET requests when JS is disabled, or just make them work normally when it's active.
He also shows examples of how to use server side DOM to apply style sheets before rendering and other cool stuff.
The demos can be found on his GitHub Account.
The part that's missing IMO to make this really awesome, is some kind of underlying storage of the widget state. So that one can visit the page without JavaScript and everything works as expected, then they turn JS on and now the widget have the same state as before but work without page reloading, then throw in some saving to the server + WebSockets to sync between multiple open browser.... and the next generation of unobtrusive and gracefully degrading ARIA's is born.
Original Answer
Well go ahead and built it yourself then.
Seriously, 90% of all WebApps out there work fine with a REST approach, of course you could do magical things like superior user tracking, tracking of downloads in real time, checking which parts of videos are being watched etc.
One problem is scalability, as soon as you have more then 1 Node process, many (but not all) of the benefits of having the data stored between requests go away, so you have to make sure that clients always hit the same process. And even then, bigger things will yet again need a database layer.
Node.js isn't the solution to everything, I'm sure people will build really great stuff in the future, but that needs some time, right now many are just porting stuff over to Node to get things going.
What (IMHO) makes Node.js so great, is the fact that it streamlines the Development process, you have to write less code, it works perfectly with JSON, you loose all that context switching.
I mainly did gaming experiments so far, but I can for sure say that there will be many cool multi player (or even MMO) things in the future, that use both HTML5 and Node.js.
Node.js is still gaining traction, it's not even near to the RoR Hype some years ago (just take a look at the Node.js tag here on SO, hardly 4-5 questions a day).
Rome (or RoR) wasn't built over night, and neither will Node.js be.
Node.js has all the potential it needs, but people are still trying things out, so I'd suggest you to join them :)
DailyJS has a good tutorial (long series of 24 posts) that walks you through all the aspects of building a notepad app (including all the possible extras).
Heres an overview of the tutorial: http://dailyjs.com/2010/11/01/node-tutorial/
And heres a link to all the posts: http://dailyjs.com/tags.html#nodepad
The Node Knockout competition wrapped up recently, and many of the submissions are available on github. The competition site doesn't appear to be working right now, but I'm sure you could Google up a few entries to check out.