Extremely, offensively rich
A correspondent to this site asked recently whether the story that they had been told about the origin of this phrase, during a visit to Victorian cemetery in London, was correct. The gist of the story was that, in the 19th century, rich people were interred in mausoleums above the ground rather than in graves like poorer people. The intention was supposed to be that the wealthy occupants of the tombs would be in a position to leave their coffins should they be interred prematurely. Being open to the air, the decomposing bodies of the wealthy occupants were inclined to smell - hence the term 'stinking rich'.
The story is entirely untrue and, once again, we have tour guides to thank for adding to the list of totally bonkers folk etymological derivations (see the Nonsense Nine). What is interesting is the large number of phrases that are popularly supposed to have derived from fears of premature burial. As well as stinking rich we also have dead-ringer, saved by the bell and bats in the belfry - none of which in reality have any connection with burial, premature or otherwise.
The real origin of stinking rich, which is a 20th-century phrase, is much more prosaic. 'Stinking' is merely an intensifier, like the 'drop-dead' of drop-dead gorgeous, the 'lead pipe' of lead pipe cinch or, more pertinent in this case, the 'stark-raving' of stark-raving mad. It has been called upon as an intensifier in other expressions, for example, 'stinking drunk' and 'we don't need no stinking badges'
The phrase's real derivation lies quite a distance from Victorian England in geography as well as in date. The earliest use of it that I can find in print is in the Montana newspaper The Independent, November 1925:
He had seen her beside the paddock. "American." Mrs Murgatroyd had said. "From New England - stinking rich".