First, we can find the prime factorization of both numbers. We have $45=3^2\cdot5$ and $15=3\cdot5$. Since the prime factorization of $45$ includes a $3^2$ term, and the prime factorization of $15$ includes only one $3$ term, the greatest common factor of $45$ and $15$ is $3$. Since $3$ is a factor of $45$ and $15$, it must also be a factor of $45^{a^2b}$ and $15ab$. Therefore, the greatest common factor of $45^{a^2b}$ and $15ab$ is $\boxed{3}$.
