Most KM's For One Country?

I assume by this question you mean a piece of real estate that has changed hands the most number of times throughout history.

I would nominate Armenia. For centuries, the area was disputed between the great powers of East (Rome) and West (Persia). Sometimes it was independent, sometimes a puppet state backed by one of the great powers, sometimes directly annexed. After the rise of Islam, it was sometimes independent, sometimes conquered. Then it became frontier territory again, this time between North (Russia) and South (Ottoman Turkey). The area known historically as "Armenia" is now divided into three zones of control: Turkish Armenia, the modern Republic of Armenia and Nagorno-Karabakh, nominally under Azerbaijan control (the latter two were formerly in the Soviet Union).

The Holy Land would also qualify as a land that has changed hands often, and for similar reasons: for millennia, it was on the frontier between the two ancient centres of civilization, Egypt and Mesopotamia. Throw in the Romans, Arabs, Crusaders, Turks and British, and you've got a very long, convoluted history of ownership.

For more modern examples of lands that changed hands often, I'd look to the Caribbean islands. These are small, and most of the time not worth anyone's while to garrison and defend properly. Thus whenever war broke out in Europe, ships would be despatched to re-annex the islands. For St Kitts, for example, the history reads: Natives, French, Spanish, Natives, English, Anglo-French Partition, Spanish, Anglo-French Partition, French, Anglo-French Partition, French, English, Anglo-French Partition, French, British. All this between 1600 and 1713.