First point - if they are appositives, they are supplements, i.e. the non-restrictive kind of appositive. I've underlined the items in question:
(1) The cat, my favorite pet's, milk bowl is too full.
(2) The cat's, my favorite pet, milk bowl is too full.
(3) The cat's, my favorite pet's, milk bowl is too full.
The simple answer is no. Supplementary (non-restrictive) appositives must of the specifying (identifying) kind, not of the ascriptive kind. The NPs in your examples are all ascriptive, so that rules them out as being appositives.
Even a specifying example like
(4) The cat's, Tiddles', milk bowl is too full.
seems clumsy and only marginally acceptable due to the two adjacent genitive NPs.