"This little book represented the culmination of 15 years' work" is correct. "This little book represented the culmination of 15 year work" is incorrect. "This little book represented the culmination of 15 years of work" is also correct. "This little book represented the culmination of a 15-year effort" is also correct. "Year" is a noun, not an adjective. "Years" is a plural noun. We are forming the possessive, which acts like an adjective, by adding an apostrophe. (Heaven only knows why an apostrophe in that position has that meaning, but it does). You can form the possessive of both singular and plural nouns. (My mom always liked to buy my clothing at the "Country Boys' and Men's Shop" and I am convinced that she liked it because they put the apostrophes in the right places.)

"This little book represented the culmination of 15 years' work." The apostrophe is required, and it goes after the "s" - the years (plural) "own" the work.

The first one is correct. Without the "s":"This little book represented the culmination of a fifteen year period of work."

You could say This little book represented the culmination of 15 years' work This little book represented the culmination of 15 year's work This little book represented the culmination of 15 years of work So, both are correct, it's just what you prefer. I've heard that other people prefer using the apostrophe before the 's' if the amount of time is not very much and after the 's' if the amount of time is a lot.