The cost of $3$ hamburgers, $5$ milk shakes, and $1$ order of fries at a certain fast food restaurant is $\$23.50$. At the same restaurant, the cost of $5$ hamburgers, $9$ milk shakes, and $1$ order of fries is $\$39.50$. What is the cost of $2$ hamburgers, $2$ milk shakes and $2$ orders of fries at this restaurant?
1. We start with the given system of equations:
   \[
   \begin{cases}
   3h + 5m + f = 23.50 \\
   5h + 9m + f = 39.50
   \end{cases}
   \]
   where \( h \) is the cost of a hamburger, \( m \) is the cost of a milkshake, and \( f \) is the cost of an order of fries.

2. Subtract the first equation from the second to eliminate \( f \):
   \[
   (5h + 9m + f) - (3h + 5m + f) = 39.50 - 23.50
   \]
   Simplifying, we get:
   \[
   2h + 4m = 16
   \]
   Dividing the entire equation by 2:
   \[
   h + 2m = 8 \quad \text{(Equation 3)}
   \]

3. Now, we have two equations:
   \[
   \begin{cases}
   3h + 5m + f = 23.50 \\
   h + 2m = 8
   \end{cases}
   \]

4. Solve Equation 3 for \( h \):
   \[
   h = 8 - 2m
   \]

5. Substitute \( h = 8 - 2m \) into the first equation:
   \[
   3(8 - 2m) + 5m + f = 23.50
   \]
   Simplify:
   \[
   24 - 6m + 5m + f = 23.50
   \]
   \[
   24 - m + f = 23.50
   \]
   \[
   f = 23.50 - 24 + m
   \]
   \[
   f = m - 0.50
   \]

6. Substitute \( h = 8 - 2m \) and \( f = m - 0.50 \) into the second equation:
   \[
   5(8 - 2m) + 9m + (m - 0.50) = 39.50
   \]
   Simplify:
   \[
   40 - 10m + 9m + m - 0.50 = 39.50
   \]
   \[
   40 - 0.50 = 39.50
   \]
   \[
   39.50 = 39.50
   \]
   This confirms that our expressions for \( h \) and \( f \) are consistent.

7. Now, we need to find the cost of \( 2 \) hamburgers, \( 2 \) milkshakes, and \( 2 \) orders of fries:
   \[
   2h + 2m + 2f
   \]
   Substitute \( h = 8 - 2m \) and \( f = m - 0.50 \):
   \[
   2(8 - 2m) + 2m + 2(m - 0.50)
   \]
   Simplify:
   \[
   16 - 4m + 2m + 2m - 1
   \]
   \[
   16 - 1
   \]
   \[
   15
   \]

The final answer is \(\boxed{15}\).