From Minkowski's Convex Body theorem it follows that a lattice can have Eucliean norm-squared of 4 or greater only in dimensions 12 or higher. So there is at least a lower bound for $n$.

Thank you Scott. I see now that inner product can be made arbitrarily close to 1.

Indeed, I forgot to divide by the length of $u$ itself. Fixed now. The only relationship between $\Gamma$ and $\mathbb{Z}^N$ is that they have the same rank. The point of my question is to understand what the above quantity can be depending on $\Gamma$.

I can see that the convergents of $\sqrt{2}$ indeed produce smaller and smaller numbers (with the observation that $| \sqrt{2} - \frac{p_n}{q_n}| < \frac{c}{q_n^2}$ for some constant c. However, I don't see the "middle steps". Namely, how does it happen that $u_1 + u_2 = p_n$ and differ by 1? Does the Pell equation come into satisfying the quadratic constraint at all? What if I change the 2 to 3, such that now I'm minimizing $(u_3 + \frac{u_1 + u_2}{\sqrt{3}})^2$? Or instead, I minimize the expression $(u_4 + \frac{u_1 + u_2+u_3}{\sqrt{2}})^2$ subject to $u_1^2+u_2^2+u_3^2 = u_4^2$?