I taught Singers' Diction in grad school for several years.
Technically, "fire" contains a triphthong, three vowels in a row, and is thus one syllable. In International Phonetic Alphabet, it's [a] then [I] then a schwa, which I can't type in ASCII.
The only other triphthong in English is the one in the word "our" or "hour", which is [a], then [U], then a schwa.