To determine the largest score the first player can achieve, we must analyze how the scores are calculated and devise a strategy for maximizing the score in any $3 \times 3$ square.

The board is a $5 \times 5$ grid, so we have several overlapping $3 \times 3$ squares to consider. When full, there are exactly nine $3 \times 3$ squares on the board. The score of the first player is the maximum sum of numbers within any of these $3 \times 3$ squares.

Consider the first player, who places a $1$ in a cell, while the second player places a $0$. The first player needs to maximize the number of $1$s in a specific $3 \times 3$ square. To achieve the maximum score, the first player should attempt to make the arrangement of $1$s dense in one area to maximize the overlap in $3 \times 3$ grids.

To strategize, note that the first player plays first, allowing them to control the initial placement of $1$s. One potential optimal strategy is:

- Fill the central $3 \times 3$ square completely with $1$s.

After filling out the entire board, count the sums in each $3 \times 3$ square. If the first player manages to place $1$s strategically, maximizing a $3 \times 3$ square's sum means achieving as many $1$s as possible within it, while the rest are filled with $0$s.

One example is placing:
- $1$s in a $2 \times 3$ or $3 \times 2$ block, ensuring the largest strategic overlap achieving maximum in any $3 \times 3$ sub-square.

In the best-case scenario (optimal placement), determining areas where all overlaps within a sub-square are maximized allows the first player to ensure six $1$s in some $3 \times 3$ square, irrespective of the opponent's placements. Each of these placements ensures substantial control over the game within limited directions and maximizes the sub-square's potential score.

Thus, the largest score the first player can ensure is:

$$\boxed{6}$$

This score of $6$ represents the maximum achievable sum of $1$s within any valid $3 \times 3$ square, accounting for strategic placements irrespective of the opponent’s moves.