(reposting as I accidentally sent AC)
There are several Linux/Android based handhelds which are exactly this. It started with the GP32 and GP2X (later Wiz / Canoo) and they gained quite a following. Today the alternatives are the NVidia Shield (bulky, but has Android on a powerful GPU) , and the planned successor of the OpenPandora (this one didn't do well due to production issues).
Problem with the above was snobs saying their PSP/NDS could do all that. Kind of true, but the difference is these are completely open platforms. No dedicated (good) first-party games though.