The progress hat is a magical, talking hat used to keep track of your current pizazz points obtained from the Mage Training Arena. It replies with a sour attitude toward your character when asked about your points.
If the hat is destroyed, points will not be lost. It is free to replace, just speak to the entrance guardian.
The hat has a maximum point storage limit, as follows:
4000 Telekinetic Pizazz Points
8000 Alchemist Pizazz Points
16000 Enchantment Pizazz Points
4000 Graveyard Pizazz Points
Also the Pizazz Progress Hat will glow brighter in order to show you how many points you have gained and how many you can still earn. There are three stages - Dull, Energised, and Full. The transition between dull and energised seems to occur at around 9,000 total pizazz points, and the final transition at around 16,000 points.