Which country has the only non-rectangular national flag?

Answer

Nepal's flag is two stacked pennants.

Context

Nepal's flag is built from two stacked triangular pennants, a form with roots in the pennants flown by Himalayan dynasties. The current design was formally adopted on 16 December 1962 under a new constitution, which laid out its construction in a step-by-step geometric recipe instead of giving only dimensions. Crimson is the national color and the blue border stands for peace; the upper pennant carries a stylized moon and the lower a twelve-pointed sun. The flag's outline is therefore defined by construction rules rather than a simple width-to-height ratio.

Sources usually explain the two points as a reference to the Himalayan peaks and the two pennants as representing the two main branches of a former ruling family, and the sun and moon as hopes that the nation will endure as long as those bodies. Interpretations vary between accounts, so the symbolism is best treated as traditional rather than documented in one definitive text.

The 'only non-rectangular flag' claim holds for national flags of sovereign states. Switzerland and Vatican City use square flags, which are rectangles with equal sides, so they do not count as exceptions in the strict sense. Ohio's state flag is a swallowtail burgee, which shows that non-rectangular flags exist below the national level, and Nepal is simply the only sovereign state to use one.