Numberlink 1

Here is one Numberlink puzzle.

This construction is easily generalized to show that $n$ clues suffice to force a unique solution on a $2^n - 1$ by $N$ grid whenever $N \ge 2^n - 1$ . That the shorter side of a grid which can be forced with $n$ clues grows at least exponentially with $n$ is quite surprising – when I first started wondering about this, I expected linear growth with a slope of about 2 or 4. Is a shorter side larger than $2^n - 1$ possible?