A popular indulgence during Mardi Gras is the King Cake, a cinnamon bread roll with a creamy frosting inside and atop the cake. It's quite tasty.
A little history...
King cakes originated in Europe in honor of the Twelfth Night (January 5th), which some mark as the true beginning of the Mardi Gras festival. The cake was usually decorated in three colors - yellow, green, and purple - celebrating the three kings that came bearing gifts (hence additional decorative coins, beads, etc.) for the birth of Jesus. Within each cake there was a trinket representing Christ, himself. These days, that trinket has popularly become a plastic baby, and the person that receives the slice of cake with the baby is named the King of Mardi Gras (which also means you are in charge of hosting next year's festivities, or at least providing the next King Cake).
In my home state of Michigan, we celebrated Fat Tuesday with punchkies (jelly-filled donuts). Delicious. Those in NOLA tend to go all out with food, drink, music, and you name it.
Whatever your indulgence may be, hurry up, as it's nearing your last chance to imbibe in all the gluttony that you can handle. Come tomorrow, it's Ash Wednesday, which marks the beginning of Lent, and while not everyone practicing Mardi Gras practices Lent, you do kind of lose your excuse to party...