Operation Abyss: New Tokyo Legacy

You can get this during the second semester but I recommend waiting until you complete the game before attempting. This is a random battle located in the Catacombs - West at X09 Y09. Getting this battle is completely luck based. I personally used the Academic's Call Portal to warp myself near this location, and used Return Gate to exit if he doesn't show up. When I got this battle, my Encounter Gauge was maxed out, though I don't know if it has any effect.