soon chromebooks will be able to run linux apps, and there guides to install LoL onto linux.
"Google built a custom virtual machine (VM), inside of which runs a version of Debian. That way, the people actually using the Chromebook don't miss out on those security features and get to play with a version of Linux that can directly integrate into the Chrome OS desktop. Google says Linux apps can be launched straight from the Launcher and can tap into files saved in your Downloads folder."
I don't think it will be too long when the day comes that we will run lesser graphic intensive games like League from our phones.
hardware wise most of the new phones today could most likely run league smoothly.
they would need to sell like a cost effective NUC-like chromebook. That you could connect your monitor mouse and keyboard to.
Edit: they already do it's called a Chromebox.