At about Level 2,147,483,647 there is an error within the game that prevents you from going deeper. This is apparently the farthest that you may go, or the "last level". However, it is doubtful that most people have enough hard drive space (or time, for that matter) to reach that level.
The reason behind the number 2,147,483,647 is the number limitation in the programming language to create Fate. Because of that, the dungeon level cannot exceed it. Nor can your currency within Fate.