Over the edge of the sofa on his fat wrist is not two but three preposition phrases, to wit
- over the edge
- of the sofa
- on his fat wrist
3 (boldfaced) prepositions, each with its own object NP, equals 3 preposition phrases.
It is certainly the case that there are frequent compound preposition constructions with words like edge; but they're no different in principle from compound noun constructions like horse doctor, pony ride, and snake bite.
- [[on/under/over/from/at] [the edge/side/back/arm]] [of the chair/sofa/table/desk]
It is also true that, while the first two PPs do go together as a single constituent in this way, the third one is independently attached to dangling, as a further locative, pinning down Dudley's watch, the antecedent of the relative clause that contains all these PPs.