Would a game that is essentially a micro Linux distro count? I feel that should be pretty doable as a bootable USB stick or CD.
If you did it that way you’d have to bundle the Linux kernel plus graphics drivers at a minimum. But I wonder how much of the OS you could avoid having. Certainly you wouldn’t need a Desktop Environment. I wonder if you would need something like X or Wayland or if you could get away without that (to run games built in a normal-ish userspace way). I guess finding the minimal environment for SDL would be a good starting point. That sounds like an interesting exercise for sure.
Although something like that probably isn’t as pure as you’re looking for, it would be pretty cool to do anyway. Maybe we should start a club.