Something that you
will eventually get and continue to get once you become a Marine. Getting the "big green weenie" may include such fine events as: getting
fucked with duty on a particularly important weekend, getting in trouble for some small petty
shit while being in garrison, getting stationed at a shithole like
29 Dumps.
The "big" represents the big bad Marine Corps picking on the little guy, the Marine. The "green" represents the true color of the Corps. The "weenie" represents getting fucked over.