Words whose plural has a plural

Solution 1:

Peoples is not the plural of the plural of person, it is the plural of the singular people. A people is a group of persons with a shared heritage, association, or trait.

The Dominicans are a proud people.

It just happens that a related meaning of people is as the plural of person, especially when referring to persons who have some identity as a group.

One similar case are for names of animals which serve as their own plurals--fish, sheep, etc. Pluralizing these can be correct, but mostly when referring to multiple categories.

My favorite fishes are the betta and the clownfish.

Solution 2:

Spaghetto, spaghetti, spaghettis.

Spaghetto,: A single strand of spaghetti.

Spaghettis: the plural of spaghetti. One usage is "spaghetti Western". Another is "spaghetti recipe", as in

This is one of the best spaghettis I have ever had.

EDIT: As Mike points out in the comments,

Panino, panini, paninis,

is a better example.