Pooh has an unlimited supply of $1\times1$, $2\times2$, $3\times3$, and $4\times4$ squares. What is the minimum number of squares he needs to use in order to fully cover a $5\times5$ with no $2$ squares overlapping?
To solve this problem, we need to cover a $5 \times 5$ square using the minimum number of smaller squares of sizes $1 \times 1$, $2 \times 2$, $3 \times 3$, and $4 \times 4$. We must ensure that no two squares overlap.

1. **Calculate the total area to be covered:**
   The area of the $5 \times 5$ square is:
   \[
   5 \times 5 = 25 \text{ square units}
   \]

2. **Consider the largest square that can fit:**
   The largest square that can fit within a $5 \times 5$ square is a $4 \times 4$ square. The area of a $4 \times 4$ square is:
   \[
   4 \times 4 = 16 \text{ square units}
   \]
   Placing a $4 \times 4$ square in one corner of the $5 \times 5$ square leaves a $1 \times 5$ strip and a $4 \times 1$ strip.

3. **Cover the remaining area:**
   - The $1 \times 5$ strip can be covered by five $1 \times 1$ squares.
   - The $4 \times 1$ strip can be covered by four $1 \times 1$ squares.

4. **Count the total number of squares used:**
   - One $4 \times 4$ square
   - Nine $1 \times 1$ squares

   Therefore, the total number of squares used is:
   \[
   1 + 9 = 10 \text{ squares}
   \]

5. **Verify if fewer squares can be used:**
   - Using smaller squares like $3 \times 3$ or $2 \times 2$ would increase the number of squares needed to cover the remaining area, as they would not fit as efficiently as the $4 \times 4$ square.

Thus, the minimum number of squares needed to cover the $5 \times 5$ square is 10.

The final answer is $\boxed{10}$