What are some verses in Shakespeare's Hamlet that speak of decay and stench?  

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The themes of decay and stench, or variations on them, occur throughout the play, beginning when Marcellus observes in the first act that "Something is rotten in the state of Denmark." Later, in a give-and-take with Polonius, Hamlet references the fact that the sun "breed maggots in a dead dog, being a good kissing carrion." In Act IV, Scene 3, the famous scene in Gertrude's chamber, Hamlet speaks of the "rank sweat of an enseamed bed, Stew'd in corruption" in reference to his his mother's union with Claudius. Still later, he engages in an extended, macabre joke about the whereabouts of Polonius's corpse, which he says is "at supper":

Not where he eats, but where he is eaten. A certain 
convocation of politic worms are e'en at him. Your 
worm is your only emperor for diet. We fat all creatures else to fat us, and we fat ourselves for maggots. Your fat king and your lean beggar is but variable service, two dishes, but to one table. That's the end.

Later he says that if they do not find his corpse within the month, they will "nose [smell] him as you go up the stair." Of course, Hamlet delivers an extended speech on the effects of death, morbidly brought to his attention by the sight of Yorick's skull. So death and decay are everywhere in Hamlet, both literally and figuratively: murder and betrayal are rotting away the moral center of the kingdom.