Last week's edition of the Radio 4 comedy programme I'm Sorry I Haven't a Clue included a jokey snippet about whether the plural of mongoose was mongooses or mongeese (listen here at 2:20 min in). The confusion comes about, of course, because of the second syllable, goose. However, the word mongoose is unrelated etymologically to the word goose. The OED tells us that mongoose is of multiple origins. It is partly a borrowing from Portuguese, and partly a borrowing from Marathi, which is a language spoken in India. The spelling mongoose came about as a result of folk-etymology. The animal has been spelled in many different ways over the centuries, including mounggoutia, mungoes and mangoust. The plural is mongooses, although the OED does list the rarer options mongeese, mongoose, and mongooze; these last three forms are described as 'irregular'.
Goose in Old English was spelled gos; its plural was ges or gees, and we are still using an adapted form of this plural today. Old English had different classes of nouns which were declined in different ways and goose belonged to a small group of nouns that formed their plural by changing the main vowel. This group also included man/men and tooth/teeth. Johnson's Dictionary of 1755 has a nice definition of goose: "A large waterfowl proverbially noted, I know not why, for foolishness". Mongoose is not in this dictionary.