Which of these would least likely result in interior damage if a built-in gutter leaks?

A built-in gutter at the eave would least likely result in interior damage if it leaks because water is directed away from the home's foundation and walls, as eaves typically extend beyond the exterior walls. This positioning helps to minimize the chances of leaking water penetrating into the interior spaces of the building.

In contrast, built-in gutters located at the ridge, valley, or around a chimney can pose a greater risk for interior damage. Gutters at the ridge can direct water towards the sides of the roof, potentially allowing it to flow downwards into the interior if leaks occur. Similarly, gutters at the valley can collect significant amounts of water flowing from different roof sections, and if they fail, water may back up and seep into the structures below. Gutters around chimneys can also face issues, as chimney penetrations create weak points where water can easily seep through, potentially causing damage to the areas around the chimney and into the home.

Thus, the least risk of interior damage comes from a built-in gutter at the eave, where water drainage is more effectively managed away from the living spaces.