First, we note that the triangle with sides $8,$ $15,$ and $17$ is a right triangle.
The longest side, $17,$ is the hypotenuse, so it is opposite the right angle.
The two longest altitudes in a triangle are the ones that intersect at the right angle and are perpendicular to the other two sides.
In a right triangle, these altitudes are simply the two legs of the triangle.
Therefore, the sum of the lengths of the two longest altitudes is $8+15=\boxed{23}$ units.The answer is: 23
Below is a math problem. Provide a detailed, step-by-step solution.

### Problem:
How many units are in the sum of the lengths of the two longest altitudes in a triangle with sides $8,$ $15,$ and $17$?

### Answer:
