Trapezium Nullis

If you want to start your own country, it's tough to find any land to lay claim to.  The entire map is painted one color or another, so unless you want to try your luck in the frigid wastes of Marie Byrd Land in Antarctica, or look for an abandoned ocean platform like Sealand, the prospects look pretty grim.

However, thanks to the wonders of international politics and border disputes, there is one patch of land that no one seems to want.  As with all border issues, the problem can be traced back to the British.   At the end of the 19th century, the British had control over the area, and they decided that the border between the region administered from Cairo and the region administered from Khartoum should be a straight line.  Any third world country which has a straight-line border drawn by a British cartographer has probably run into issues with it, and in this case the British themselves found a problem.  There's a triangle of land along the coast of the Red Sea which is closer to Khartoum and to Cairo, but was North of their line.

So, 3 years later, they drew a new line, this one dipped a bit below the old line, and then rose back above it to place that triangle under Khartoum's authority.  The area in that little dip is called the Bir Tawil Triangle (despite being roughly a trapezoid).  Fast forward to today, and both countries want the valuable Red Sea coast line, and couldn't care less about Bir Tawil.  Egypt claims the original straight line border, which places the coast line on their side, and gives Bir Tawil to Sudan, while Sudan claims the revised border, which does the opposite.  Both sides vehemently insist that Bir Tawil does not belong to them (as accepting it would mean ceding the much more valuable territory), and it is therefore essentially the last unclaimed habitable territory in the world.