As you play through the main game, you'll sometimes find letters in crates or as drops from enemies. Once you have nine of these spelling out "WHEREAMI?" on the left side of the screen you'll get a ticket for another tower, Transdimensional Leakage, which you can access by choosing Hard when you leave the lodge.
This new tower starts off at about the same difficulty as the main tower ends, and you have no continues, but once you make it past the first 100 floors you get to the first boss, Horror. He's actually really simple - he appears, shoots one spike ball at you, then disappears. He'll do it twice from high up and then once from ground level, then repeat. Lightning magic when he's high or projectiles when he's low both work great. Once he's dead the achievement will unlock.
Note: The tickets to Transdimensional Leakage are consumed only when you choose that tower in the lodge, not once per death. As long as you don't quit you can attempt it as many times as you like on just the one ticket.