Initially, I was tinkering around a lot with x11vnc and the dummy-X11 Server to get a nice persistent Desktop you could also move to other desktops and see at multiple places.
I also used various techniques to successfully run Linux in a chroot or proot environemnt on my rooted Android phone. My student Dominik Költringer (pdf download) wrote a bachelor thesis on this topic. I am still thinking about building a more mature prototype and trying a crowdfunding approach to see, how much interest there is in being able to easily carry your Desktop and all your data around with you. Phones are getting more and more powerful and ideas like Microsoft's Continuum and Ubuntu Phone are pointing in the same direction. Another very interesting approach is Andromium.
If you like to discuss these ideas and share results, just contact me.