jmc

At the end of the first mile, there will be $\frac{2}{3}$ of the initial milk in the bucket. Each additional mile multiplies this amount by $\frac{2}{3}$. Thus, when he arrives at home at the end of the third mile, there will be $\frac{2}{3}\cdot \frac{2}{3}\cdot \frac{2}{3}={\left(\frac{2}{3}\right)}^3$ as much milk in the bucket. Since he had 2 gallons initially, the amount in the bucket when he gets home is $2\cdot {\left(\frac{2}{3}\right)}^3$. Because ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$, this expression is equivalent to $2\cdot \frac{2^3}{3^3}$. Because $n^a\cdot n^b=n^{a+b}$, this equals $\frac{2^4}{3^3}$. Multiplying the exponents out, we get $\boxed{\frac{16}{27}}$ gallons.