Largesse is extreme generosity. If your neighbors bring you an expensive watch from Switzerland because you fed their cat while they were traveling, thank them for their largesse.
Though pronounced "lar JESS," the word largesse looks sort of like largeness. That's an easy way to remember what largesse means: think of it as largeness of spirit. Largesse can describe the generosity of someone giving gifts, or it can describe the gift itself. You could dispense largess by buying your brother a dinner at a fancy restaurant for helping you out.