 Solution

Take the degree-$d$ Chebyshev polynomial $T_d$, characterized byIts Julia set is the interval $[-1,1]$. The complement $\hat{\mathbb{C}}\setminus [-1,1]$ is connected, and it is exactly the basin of infinity. Hence $T_d$ has exactly one Fatou component. This is the Chebyshev polynomial Julia set example.