We are given a $4\times4$ square, consisting of $16$ squares with side length of $1$. Every $1\times1$ square inside the square has a non-negative integer entry such that the sum of any five squares that can be covered with the figures down below (the figures can be moved and rotated) equals $5$. What is the greatest number of different numbers that can be used to cover the square?


1. **Understanding the Problem:**
   We are given a $4 \times 4$ grid where each cell contains a non-negative integer. We need to ensure that the sum of any five cells that can be covered by the given figures equals 5. The figures can be moved and rotated.

2. **Analyzing the Figures:**
   The figures are not explicitly given, but we can infer that they cover five cells in some configuration. Common configurations include:
   - A cross shape (like a plus sign)
   - An L shape
   - A straight line of five cells

3. **Setting Up the Grid:**
   Let's denote the cells of the $4 \times 4$ grid as follows:
   \[
   \begin{array}{cccc}
   a_{11} & a_{12} & a_{13} & a_{14} \\
   a_{21} & a_{22} & a_{23} & a_{24} \\
   a_{31} & a_{32} & a_{33} & a_{34} \\
   a_{41} & a_{42} & a_{43} & a_{44} \\
   \end{array}
   \]

4. **Placing the Figures:**
   Consider the cross shape figure. If we place it at the center of the grid, it covers the cells $a_{22}, a_{21}, a_{23}, a_{12}, a_{32}$. The sum of these cells must be 5:
   \[
   a_{22} + a_{21} + a_{23} + a_{12} + a_{32} = 5
   \]

5. **Generalizing the Placement:**
   By moving and rotating the figure, we can cover different sets of five cells. For example:
   - Placing the cross at $a_{33}$ covers $a_{33}, a_{32}, a_{34}, a_{23}, a_{43}$.
   - Placing the cross at $a_{11}$ covers $a_{11}, a_{12}, a_{21}, a_{22}, a_{13}$.

6. **Finding the Pattern:**
   To satisfy the condition for all possible placements, we need to ensure that the sum of any five cells covered by the figure is always 5. This implies a certain symmetry and repetition in the values of the cells.

7. **Constructing the Grid:**
   Let's try to construct a grid that satisfies the condition. We can start with a simple pattern and check if it works:
   \[
   \begin{array}{cccc}
   1 & 1 & 1 & 1 \\
   1 & a & b & 1 \\
   1 & b & a & 1 \\
   1 & 1 & 1 & 1 \\
   \end{array}
   \]
   Here, the sum of any five cells covered by the cross figure should be 5.

8. **Checking the Values:**
   - For the center cross: $1 + a + b + 1 + 1 = 5 \implies a + b + 3 = 5 \implies a + b = 2$
   - For other placements, similar equations will hold.

9. **Maximizing the Number of Different Numbers:**
   To maximize the number of different numbers, we can choose $a$ and $b$ such that they are distinct and non-negative integers. The simplest choice is $a = 2$ and $b = 0$ (or vice versa).

10. **Conclusion:**
   The greatest number of different numbers that can be used to cover the square is 3 (the numbers 0, 1, and 2).

The final answer is $\boxed{3}$