Absolutely, and the init system does not even have to set up the filesystem and all. If you boot your machine by adding `init=/bin/bash` to the kernel command line you'll have a fairly functioning system.

Do anything necessary from there to boot your game, and record those steps in a script. When that's done you can just point your init cmdline to that script (doesn't even have to be a binary, a script with the #!/bin/bash shebang should just work).