Ouya on a PC monitor.
I know I'm not the only person to try this, but I did manage to succeed in getting it working. I have just a 20 inch lcd monitor, with a native res of 1600x900. Its a fairly decent screen, but it doesn't have HDMI, just DVI and VGA. Plugging in an HDMI to DVI adapter got me a picture, in 640x480. And I still didn't have a way to get sound out of it. Now I like to make things complicated, so I did some research into modifying the screens EDID table. That never really panned out. But while I was toying with all that I ordered an HDMI to HDMI+sound adapter.
When it arrived it was the complete solution to my problem. The adapter doesn't list it on its features, but it does have EDID masking. It caused the Ouya to output 1080p. With that, there was nothing on screen. But I had already installed xposed and the mod collection, So in 640x480 I set 720p from the collection and rebooted. Its working perfectly now, and I've got a cheap game and media station at the foot of my bed.
I bought the HDMI sound adapter from here.
It doesn't have any latency I've noticed and doesn't require any power. Probably the cheapest solution to this problem too, I pinched quite a few pennies on this. I came out of this spending maybe $90, as I bought the Ouya from the promo sale that had it at $60.