To indulge in something means "to yield to the desires and whims of, especially to an excessive degree". This neatly describes the major attitude toward life of most of the characters in The Great Gatsby. They lived to enjoy what they considered to be the finer things and experiences of life to excess.
Gatsby's entertaining style was marked by the indulgences with which he strove to impress his guests.
Every Friday five crates of oranges and lemons arrived from a fruiterer in New York - every Monday these same oranges and lemons left his back door in a pyramid of pulpless halves...At least once a fortnight a crop of caterers came down with several hundred feet of canvas and enough colored lights to make a Christmas tree of Gatsby's enormous garden.
On a different scale, Tom Buchanan indulged the wishes of his mistress, Myrtle Wilson. After being approached by a man attempting to sell "a dozen very recent puppies of an indeterminate breed" and hearing Myrtle express her desire to have one, the purchase was quickly completed, even though Tom suspected the puppies were overpriced.
Tom and Daisy also lived a life of indulgence. They fulfilled their individual and joint pleasures without consideration for the cost or consequences of their actions. Nick's conclusion regarding their lifestyle was that "they smashed up things and creatures and then retreated back into their money or their vast carelessness...and let other people clean up the mess they had made"
All the characters partake in indulgence in The Great Gatsby. Tom indulges in an illicit affair with Myrtle where he exploits her and uses his high status to shy away from consequence. Gatsby indulges by trying rekindle his romance with Daisy. Daisy in indulgent in that she enjoys Gatsby's advances even though she has no true intention of leaving Tom and her daughter. The residents of East and West egg indulge in their own desires and give little to no though for others.