How many syllables are there for "laboratory"

The transcription /ˈlabrəˌtôrē/ is correct (for American English; and be aware that it is a non-IPA transcription), and you have interpreted it and the recording correctly. The word is often pronounced with four syllables. Unstressed syllables in certain contexts are often dropped from the middle of words.

A five-syllable pronunciation, /ˈlabərəˌtôrē/, seems theoretically possible to me, because of the spelling, but I don't know if I've ever heard it used. It is not common, and might sound over-enunciated.

There are many other examples of words that can elide unstressed vowels after stressed syllables and before resonant consonants, like family and chocolate.

In British English, the word usually is pronounced with a different stress pattern and elision affects a different vowel. The Oxford English Dictionary gives "/ləˈbɒrət(ə)ri/", which as you can see may (and generally does) elide the vowel before the second /r/.

This is related to a more general pattern where American English speakers have secondary stress on the penult syllable of words ending in -ary and -ory, and some words ending in -ery (like cemetery) while British English speakers do not have secondary stress here, and so can elide these vowels (leading to pronunciations like diction'ry, regulat'ry, cemet'ry).