Because seem like someone who might have an interesting answer:

A GeoGuessr player, GeoWizard, has done a few “straight line challenges”, where he attempts to walk across a country in as straight a line as possible, usually planning beforehand with Google Earth and PostGIS. This got me thinking of what could fairly be thought of as “crossing”, since obviously you couldn’t describe e.g. walking from one side of Florida to the other as “crossing the USA”.

My best thought was to set the ending point of the line by following the border of the country in each direction til they met on the other side. To avoid the fractal coastline problem, use the challenger’s stride length as the unit of measure for the border.

But perhaps there is a better, more rigorous way of defining the opposite point on the edge of an arbitrary polygon.

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 ).