7. Remove any $2 \times 2$ small square from the corner of a $3 \times 3$ grid, and the remaining shape is called a "corner shape" (Figure 3 is an example of a corner shape). Now, place some non-overlapping corner shapes in a $10 \times 10$ grid (Figure 4). The boundaries of the corner shapes must coincide with the boundaries or grid lines of the grid. Find the maximum value of the 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.
7. First, $k_{\max }$
$<8$. This is because, if eight corner shapes are placed in the manner shown in Figure 9, it is impossible to place another corner shape in the grid.

Next, we prove that after placing seven corner shapes arbitrarily, it is still possible to place another complete corner shape.

Cover the 5th and 6th rows and the 5th and 6th columns of the $10 \times 10$ grid, leaving four $4 \times 4$ smaller grids. After placing seven corner shapes, since each corner shape cannot intersect with two of the aforementioned $4 \times 4$ smaller grids, according to the pigeonhole principle, there must exist a $4 \times 4$ smaller grid $S$ such that at most one corner shape intersects with $S$. Since a corner shape can be contained within a $3 \times 3$ smaller grid, the part of the smaller grid $S$ occupied by the corner shape must be contained within a $3 \times 3$ smaller grid at one of its corners.

As shown in Figure 10, a new corner shape can be placed in the remaining part of the smaller grid $S$ after removing a $3 \times 3$ smaller grid from one of its corners. Therefore, $k=7$ satisfies the condition.
In conclusion, $k_{\max }=7$.