We can construct a connected plane graph $G_n$ such that the greedy algorithm (with a suitable vertex order produces a colouring using $n$ colours, and such that tall vertices are adjacent to the unbounded face.
This is clear for $n=1$. Assume $n>1$ and we already have $G_{n-1}$. Let $G_n$ consist of $n-1$ separate copies of $G_{n-1}$ and an additional vertex $v$. For $1\le i\le n-1$, add an edge from $v$ to a vertex of the $i$th copy of $G_{n-1}$ that is assigned colour $i$ by greedy colouring. On this $G_n$, perform the greedy algorithm by first running over the copies of $G_{n-1}$ in the order defined for them, and finally visit $v$. This will lead to an $n$-colouring of $G_n$. Also, all vertices of $G_n$ are adjacent to the unbounded face.