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.