Let $n$ be a positive integer.  In $n$-dimensional space, consider the $2^n$ points whose coordinates are all $\pm 1$.  Imagine placing an $n$-dimensional ball of radius 1 centered at each of these $2^n$ points.  Let $B_n$ be the largest $n$-dimensional ball centered at the origin that does not intersect the interior of any of the original $2^n$ balls.  What is the smallest value of $n$ such that $B_n$ contains a point with a coordinate greater than 2?
1. **Identify the coordinates of the points and the distance from the origin:**
   The $2^n$ points in $n$-dimensional space have coordinates $(\pm 1, \pm 1, \ldots, \pm 1)$. Each of these points is at a distance of $\sqrt{n}$ from the origin, as calculated by the Euclidean distance formula:
   \[
   \text{Distance} = \sqrt{1^2 + 1^2 + \cdots + 1^2} = \sqrt{n}
   \]

2. **Determine the radius of the ball centered at the origin:**
   We need to find the largest $n$-dimensional ball $B_n$ centered at the origin that does not intersect the interior of any of the $2^n$ balls of radius 1 centered at the points $(\pm 1, \pm 1, \ldots, \pm 1)$. For $B_n$ to be tangent to these balls, the radius of $B_n$ plus the radius of any of the $2^n$ balls must equal the distance from the origin to any of these points:
   \[
   r + 1 = \sqrt{n}
   \]
   Solving for $r$, we get:
   \[
   r = \sqrt{n} - 1
   \]

3. **Determine the condition for $B_n$ to contain a point with a coordinate greater than 2:**
   We need $B_n$ to contain a point with at least one coordinate greater than 2. The farthest point from the origin in $B_n$ is at a distance of $r$ from the origin. For a point to have a coordinate greater than 2, the radius $r$ must be at least 2:
   \[
   \sqrt{n} - 1 \geq 2
   \]
   Solving for $n$, we get:
   \[
   \sqrt{n} \geq 3
   \]
   Squaring both sides, we obtain:
   \[
   n \geq 9
   \]

4. **Verify the smallest value of $n$:**
   For $n = 9$, we have:
   \[
   r = \sqrt{9} - 1 = 3 - 1 = 2
   \]
   This means that $B_9$ is tangent to the balls of radius 1 but does not intersect their interiors. However, the problem asks for the smallest $n$ such that $B_n$ contains a point with a coordinate greater than 2. Since $r = 2$ for $n = 9$, it does not strictly contain a point with a coordinate greater than 2.

   For $n = 10$, we have:
   \[
   r = \sqrt{10} - 1 \approx 3.16 - 1 = 2.16
   \]
   This radius is greater than 2, ensuring that $B_{10}$ contains a point with a coordinate greater than 2.

Conclusion:
\[
n \geq 10
\]

The final answer is $\boxed{10}$