How many cups are there in the underworld in kingdom hearts 2?

The cups you can play by asking Panic are: Pain and Panic Cup, Cerberus Cup, Titan Cup, and Goddess of Fate Cup.

Once you have cleared all of these cups, you unlock the Paradox Cups. Go to Hades in his special little chamber and talk to him and you can play the Paradox Pain and Panic Cup, Cerberus Cup, Titan Cup, and Hades Cup.

Note: Even though some of the cups have the same names, they are NOT the same cups. Paradox cups are much more difficult.