For which of the following $ n$, $ n\times n$ chessboard cannot be covered using at most one unit square piece and many L-shaped pieces (an L-shaped piece is a 2x2 piece with one square removed)?

$\textbf{(A)}\ 96  \qquad\textbf{(B)}\ 97  \qquad\textbf{(C)}\ 98  \qquad\textbf{(D)}\ 99  \qquad\textbf{(E)}\ 100$
1. **Understanding the problem**: We need to determine for which value of \( n \), an \( n \times n \) chessboard cannot be covered using at most one unit square piece and many L-shaped pieces. An L-shaped piece is a 2x2 piece with one square removed, thus covering 3 squares.

2. **Analyzing the L-shaped piece**: Each L-shaped piece covers exactly 3 squares. If we use \( k \) L-shaped pieces, they will cover \( 3k \) squares. If we use one additional unit square piece, it will cover 1 square. Therefore, the total number of squares covered by \( k \) L-shaped pieces and one unit square piece is \( 3k + 1 \).

3. **Total squares on the chessboard**: An \( n \times n \) chessboard has \( n^2 \) squares.

4. **Condition for covering the chessboard**: For the chessboard to be completely covered by \( k \) L-shaped pieces and at most one unit square piece, the total number of squares \( n^2 \) must be expressible as \( 3k \) or \( 3k + 1 \). This means:
   \[
   n^2 \equiv 0 \pmod{3} \quad \text{or} \quad n^2 \equiv 1 \pmod{3}
   \]

5. **Possible values of \( n \) modulo 3**:
   - If \( n \equiv 0 \pmod{3} \), then \( n^2 \equiv 0 \pmod{3} \).
   - If \( n \equiv 1 \pmod{3} \), then \( n^2 \equiv 1 \pmod{3} \).
   - If \( n \equiv 2 \pmod{3} \), then \( n^2 \equiv 4 \equiv 1 \pmod{3} \).

6. **Checking the given options**:
   - \( n = 96 \): \( 96 \equiv 0 \pmod{3} \) (satisfies the condition)
   - \( n = 97 \): \( 97 \equiv 1 \pmod{3} \) (satisfies the condition)
   - \( n = 98 \): \( 98 \equiv 2 \pmod{3} \) (does not satisfy the condition)
   - \( n = 99 \): \( 99 \equiv 0 \pmod{3} \) (satisfies the condition)
   - \( n = 100 \): \( 100 \equiv 1 \pmod{3} \) (satisfies the condition)

7. **Conclusion**: The only value of \( n \) for which the \( n \times n \) chessboard cannot be covered using at most one unit square piece and many L-shaped pieces is \( n = 98 \).

The final answer is \(\boxed{98}\).