Someone who is self-indulgent gives themselves a lot of treats. Parents who are indulgent cave to every desire their child expresses. Indulgent means lenient, or overly generous.
Indulgent is a word that, here in Puritanical North America, is hard to know how to take. Is it okay to “indulge yourself” as so many spa advertisements suggest? Or is indulgent always associated with excess? You can indulge fantasies of figuring out the answer, or you can give in to an indulgent shrug and move on to another word.