iPhone 6 Plus ($619)
It is only with an Apple phone that you can access iOS-only games like The Room Two, and this is the one to pick. You get a great 5.5-inch screen with 1080 x 1920 resolution and a slim build for comfort. Furthermore, the A8 64-bit processor means performance matches image quality. On the downside, it is very costly.
The Note 4 is an excellent gaming phone. I haven't found the weight to be an issue, but then again I don't spend hours holding my phone lol