Raspberry PI without OS on it?
Run code on the RaPi without OS: No problem. David Welch has done the grunt work, check his gitub. Basically, you can take the standard startup files and replace the kernel.img with the file you want to run. To avoid the 'SD card dance' you can run a bootloader that receives the image-to-be-run over a serial line. I have added an auto-reboot, so you can run a new image 'hands-off'. If you want to go this route I have some more information.

Run your own bare-metal application and use the screen and SD card: That's a problem. All information is available deep in the Linux sources, and there are persons working on this (check DexOs?), so I assume in a half a year or so this info will be common knowledge. But for now I would say this is probably too much work.