“Stinking rich” means extremely, offensively, disgustingly rich.
The word “stink,” has Germanic roots, and did not originally indicate an unpleasant aroma. In Old English, it meant to produce an odor of any kind, pleasant as well as unpleasant. Soon, however, “stink” narrowed to mean “to give off a strong offensive smell,” and by the 1200s “stink” took on the figurative meaning of simply “to be very offensive,” a sense still in use in such phrases as “My job stinks.”
The Oxford English Dictionary tells us that “stinking” became a “vague epithet connoting disgust and contempt” in the 1200s, when one person might speak of another’s “stinking pride.” But it wasn’t until the 1800s that “stinking” became an intensifier meaning “offensively” in phrases such as “stinking drunk.”
A theory exists that the phrase arose from medieval burial of rich people above ground, with the resulting horrible miasma. But that is not true.
“Stinking rich” is a 20th-century phrase. “Stinking,” as stated above, is merely an intensifier, like the “drop-dead” of drop-dead gorgeous, the “lead pipe” of lead pipe cinch or the “stark-raving” of stark-raving mad. It has been called upon as an intensifier in other expressions, for example, “We don’t need no stinking badges.”
The earliest use of it in print appears to be from the Montana newspaper The Independent, November 1925: “He had seen her beside the paddock. ‘American,’ Mrs Murgatroyd had said. ‘From New England – stinking rich’.”
Another phrase used as an intensifier is “filthy” and I’ve heard “filthy rich” many times.
When I was a child, I put a little perfume on my wrist and my mother said, “You stink pretty.” But I have a feeling she was being facetious.