Epic is a type of Mission, it will appear every few days on the Headquarters page once you have 10 recruits under your command. In an Epic Mission, your main army is assisted by your best recruits. The amount of Troopers on your team caps at 32. Together, you fight an assortment of non-playable troopers which usually outnumber yours by a notch and can carry more than 3 pieces of Equipment, but lack in levels after the first couple of Waves.
As you complete each Epic mission, the enemy count is increase the next time you attempt to complete it.