Basically since the Chromecast was released.

The Chromecast came out on July 24, 2013, and several dozen articles note the feature before the end of July (26-28th). Just click the little down arrow on the Chromecast button (part of the Chromecast extension) then "Cast entire screen (experimental)."

July 26th, 2013: http://www.droid-life.com/2013/07/26/tip-chromecast-can-cast...

July 26th, 2013: http://www.androidauthority.com/chromecast-share-entire-desk...

July 27th, 2013: http://forum.xda-developers.com/showthread.php?t=2380302

July 26th, 2013: http://www.reddit.com/r/Android/comments/1j3sd1/chromecast_c...

Oh my god, I can't believe I missed that...

Does this mean I can stream any app and my desktop or just a full screen chrome window?

Unfortunately (at least for me) it seems like it doesn't work via LAN, but somehow uses Internet connection to stream to somewhere and then back, kinda sucks and slow

Full screen streaming works beautifully for me, it's how my wife and I watch Amazon Prime movies. It's just like watching TV. I've read that the computer needs to encode the video it's streaming to the chromecast, so if your computer is slow, the streaming will be slow. That may be why you were getting poor results.

You can stream any app and your desktop, yes.