I have not rooted. One reason why I left Android back in the day (Motorola Droid 1, HTC Thunderbolt) for iPhone was because I had to CONSTANTLY root again and again to fix a ton of small problems which usually caused more problems - I just wanted a phone. After the iPhone 5 I saw that Android 4.4 was an attempt at a PURE ASOP Android experience. So I got a Nexus 7 (2013) and LOVED it so much I got a Moto X. The ability to Turn Off apps I didn't like was heaven.
Now, yeah, there's a few adware apps I can't turn off on the Note 3 but they don't bug me and they stay off unless I open them. The OS is smooth and after a few days of configuration and turning off some 15+ apps - I have the phone right where I want it. I haven't been able to test the battery fully yet as today was a holiday for my work (got Note 3 2 days ago) - will test battery tomorrow. But the battery usage I do see is 2x what I got on my Moto X.
I hear that there's a flash trip wire in the Note 3 that reports back when you root the device - making it so they know if you've rooted. Some roots claim to leave that trip wire untouched but ... I like having the security of knowing I can return it if I need to. That and, after 2+ years of rooting almost every week back with the Motorola Droid 1 and HTC Thunderbolt - I'd rather just use my phone. I'm getting good battery, I am actually enjoying TouchWiz?, and some of the Samsung apps are very useful.
I am still open to rooting, especially, as I've seen, some claim 1.5x battery life. But because I've come from the Moto X where overnight it would use 20% battery (Note 3 used 4% battery overnight last 2 nights) ..... I don't see the need - and I think that's the only reason why I'd root - to increase battery life and uninstall adware (which makes me very mad).
Sorry it's long but I REALLY love this phone.
I'll unplug it tomorrow morning and give you some real world usage tomorrow night. - Making a Google Keep note now.