Ages with product 36 and a unique eldest

Three children have whole-number ages whose product is $36$. Someone told the sum of the ages still cannot work them out, but on hearing that the eldest child is unique, they can. What are the ages?

2, 2 and 9. Of the factorisations of $36$ into three ages, only $\{1,6,6\}$ and $\{2,2,9\}$ share a sum ($13$), which explains why the sum was not enough. A single eldest rules out the twin sixes.