Which is the smallest natural number that has exactly 15 divisors?

15 = 15 or $3\times 5$, so the number is $p^{14}$ or $p^4{q}^2$. The smallest choice is $2^4\times 3^2=144$ (much less than $2^{14}$). 196 has only 9 divisors; 324 and 400 also have 15, but they are larger.