Here is the recipe for the next word:
A word length of which is 10 and uses only 4 letters (in mixed order), where one of the letters is used once, next is used twice, another is used three times, and the last is used 4 times.
Notice that the first word "A" word is just a word with single letter.
The second word - "ILL" is a word which is consisted of two letters, and one of them ("I") is used once, another one ("L") is used twice.
The third word - "COCOON" is consisted of three letters, one of which ("N") is used once, another is used twice ("C"), and the other ("O") is used three times.
So the next word should be a word which uses 4 letters, where one of the letters is used once, next is used twice, another is used three times, and the last is used 4 times.
Here is the list of the 10 letter words, where you can search the word
One of the possible words found by @HughMeyers
Also another pack of words, thanks to @Sp3000:
DEADHEADED PEPPERTREE POOPSCOOPS REMEMBERER SERENENESS SLEEVELESS SUSSARARAS SUSURRUSES TERREVERTE