Short version:
I have a Moto G 4G edition that can cast screen to chromecast without root, hacks or the chromecast app, as long as it stays at factory version of everything: android 4.4.3 and google play services 4.3.25.
Now, the long version:
I just bought a new Moto G phone, 8GB single SIM with 4G/LTE, Brazilian edition and, to my surprize, the cast screen option shows up and works every time, without any hacks, not root, not even the chromecast app needs to be installed.
I found it interesting because Moto G is not one of the models that Google officialy supported for the cast screen to chromecast feature at first, and I noticed that after a Google Play Services update, the "feature" was gone, like it "should" be.
So, I investigated a litte to document why I think it works for this phone right now. Maybe this information can be useful for others.
The cast screen options only shows if the google play services is the one that came with the phone, version 4.3.25. As of today, Google+ and Hangouts request it to be updated to version 5.0.84 to work. At this version, the cast screen option is not shown and if acessed by going to "Settings, Display, Cast Screen", no chromecast devices are detected.
I have other Moto G, the older one, 16GB dual SIM, already updated to android 4.4.4 that can't cast screen. I think that it is because of the minimun version of google play services that is newer than the one of version 4.4.3.
So, anyone out there wants / can test this version of google play services to confirm my theory? I don't know if the apk can be installed alone but I extracted it from the phone I have the cast screen working: https://drive.google.com/file/d/0B2GrmBw0Jk8_RjFDcTcwUGpqZEdJbkdOT18wZjhWTmo2aDF3/edit?usp=sharing