When a destroyer needs more than a five-inch gun

What is the largest gun fitted to a destroyer, and when does a larger calibre make sense?

Some destroyers experimented with larger guns, but a weapon above five inches needs a clear purpose because it imposes major design costs on a destroyer-sized hull.

Some designs pursued 150 mm or larger guns while remaining nominally destroyers. A World War II destroyer pays heavily for a gun above five inches, so the intended mission must justify the trade-off.