From what I understand, hour, fire, hire, layer, rhythm, etc., are all examples of words which are not easily classifiable. But, according to this linguist ^[1],

Hour and fire are generally considered to be monosyllabic words containing a triphthong.

Wikipedia further confirms this in a couple of its articles.

Triphthong (WP) ^[2]:

English in British Received Pronunciation (monosyllabic triphthongs with R are optionally distinguished from sequences with disyllabic realizations)

[aʊ̯ə̯] as in hour (compare with disyllabic "plougher" [aʊ̯.ə])

(British) Received Pronunciation ^[3]:

RP also possesses the triphthongs /aɪə/ as in ire, /aʊə/ as in hour, /əʊə/ as in lower, /eɪə/ as in layer and /ɔɪə/ as in loyal. There are different possible realisations of these items: in slow, careful speech they may be pronounced as a two-syllable triphthong with three distinct vowel qualities in succession, or as a monosyllabic triphthong. In more casual speech the middle vowel may be considerably reduced, by a process known as smoothing, and in an extreme form of this process the triphthong may even be reduced to a single vowel, though this is rare, and almost never found in the case of /ɔɪə/[47] . In such a case the difference between /aʊə/, /aɪə/, and /ɑː/ may be neutralised with all three units realised as [ɑː] or [äː].

All that said, I suspect that the most accurate answer ultimately depends on how you pronounce it yourself. Nice question :)

Not that everything I learned in school was true, but I remember being taught that most dictionaries – at least print dictionaries – broke words into syllables, and this was one of those things that dictionaries were useful for.

So, one could always count the number of •'s, and add one, and get the number of syllables in a word:

One dictionary's own definition of syllable mentions there are three in inferno, and two in water; as could be expected, the entries for those words show them being broken up that way. By the same token, the word word has one:

However, this method for counting syllables creates a few anomalies, whereby some one- and two-syllable words are pronounced very much alike (in other words, hearing the two words would lead someone to think they have the same number of syllables, but the dictionary would indicate otherwise):

(The oddest of all might be hire and higher, which I would pronounce as true heterographs, but the dictionary would break into one and two syllables, respectively.)

I found it interesting that the word syllabification was defined as the division of words into syllables, either in speech or in writing. That "either in speech or in writing" part made me wonder if one could accurately say that the word hour has one syllable in writing, and two in speech – that is, if hour was pronounced as rhyming with tower or power, and not as a heterograph with are, as some drawlers might be inclined to do.

In the end, I sup•pose it all de•pends on how you de•fine syl•la•ble, and how you de•cide to come up with an of•fi•cial an•swer in ca•ses where the ex•act num•ber of syl•la•bles is not im•me•di•ate•ly ev•i•dent or ap•par•ent.

People who are born to power
Lack the time to spare an hour
Smelling rose or other flower,
Be it sweet or be it sour.

Since those lines of trochaic tetrameter all rhyme with each other on the last foot, hour needs must contain two syllables. The stanza is 8/8/8/8 by syllables, with rhyme scheme AAAA.

Here is a different illustration of the same thing:

Tapered candles need no power,
‭    ‭    flicker though they may.
They will burn for just an hour,
‭    ‭    turning night to day.

That one has an 8/5/8/5 syllable pattern and an ABAB rhyme scheme. Unless power and hour rhyme, and are each two syllables, that doesn’t work.

IPA isn't that difficult to decipher, is it?

hour ^[1] /ou(ə)r/ (one syllable)

hour ^[2] /aʊər, ˈaʊər/ (one or two syllables, one being preferred)

I would pronounce this with two syllables:

Ow - Err

Ymmv

As long as "ou" is a diphthong, this word has one syllable.