Three. (50 points) Remove a $2 \times 2$ small square from any corner of a $3 \times 3$ grid, and the remaining shape is called a "corner shape". Now, place some non-overlapping corner shapes in a $9 \times 9$ grid, with the requirement that the boundaries of the corner shapes coincide with the boundaries or grid lines of the grid. Find the maximum positive integer $k$ such that, no matter how $k$ corner shapes are placed, it is always possible to place one more complete corner shape in the grid.
Three, first, $k_{\max }<6$. This is because, if 6 corner shapes are placed in the manner shown in Figure 4, it is impossible to place another complete corner shape on this grid.

Now, place 5 corner shapes in any manner.
Next, we will prove that it is still possible to place another complete corner shape.

Consider the five shaded areas $A, B, C, D, E$ in Figure 5. If there exists a region that does not intersect with any of the 5 placed corner shapes, then it is clearly possible to place a complete corner shape in that region.

Assume that all five regions contain a part of a corner shape. Note that a corner shape cannot intersect with two of the regions $A, B, C, D, E$ simultaneously. Therefore, each of the five regions $A, B, C, D, E$ corresponds to a unique corner shape that intersects with it, which we denote as $f(A), f(B), f(C), f(D), f(E)$.
According to Figure 5, the four $4 \times 4$ sub-grids in the corners of the $9 \times 9$ grid are labeled as $a, b, c, d$.

Note that the corner shape $f(E)$ cannot intersect with all four regions $a, b, c, d$ simultaneously. Without loss of generality, assume $f(E)$ does not intersect with $a$. Considering that the corner shapes $f(B), f(C), f(D)$ also do not intersect with $a$, the only corner shape that intersects with $a$ is $f(A)$. However, the part of region $a$ occupied by the corner shape $f(A)$ must be contained within a $3 \times 3$ sub-grid in one of the corners of $a$. Without loss of generality, assume it is contained in the shaded area shown in Figure 6, then it is possible to place a new corner shape in the remaining part of $a$ as shown in Figure 6.
The above demonstrates that $k=5$ satisfies the problem's requirements.
In conclusion, $k_{\max }=5$.