As an abstract geometric problem the greatest width of an abstract polygon in a Euclidean 2D plane is found be looking at the greatest distance between all pairs of parallel lines that have been pulled together to clamp the polygon. The maximal diameter as opposed to the minimal waist.

Some might then say that "crossing" that polygon is to travel that longest line across the greatest width.

This simplistically avoids the question of concave polygons, complex polgons with exclusions (the Vatican state is removed from the Italian contry bounds), polgon collections (the nation of Fiji has many islands and can be tricky to traverse on foot .. not forgetting that perhaps the longest diameter might be from one island to another with no other islands between).

There's also the challenge of parallel lines on a 2D 'spherical' manifold, the almost spherical abstract ellipsoid of earth (or very non ellipsoidal Geoid if we take a constant gravitation value as the surface). On such manifolds lines are Great Circles (more or less) and always intersect.

Still, lets just say you're looking for the longest walkable(?) great circle path across a country that might go outside that country and perhaps is best travelled by a crop duster at 80m ground clearance to avoid getting feet wet.

The challenge itself takes some posing.

Meanwhile, less abstractly, I do like a jolly that "crosses a country" in a manner accepted by a (Wo)Man on a Clapham omnibus.

eg: https://en.wikipedia.org/wiki/Robyn_Davidson only went "half way", but that was accepted as an epic crossing. https://thelongridersguild.com/stories/stef-gebbie.htm "only" crossed most of the E-W distance across the lower portion of the country, while https://www.abc.net.au/news/2021-04-03/french-woman-conquers... travelled North - South, the long bit, but not quite coast to coast ( https://en.wikipedia.org/wiki/Bicentennial_National_Trail ).