1. Angle InletIn far southeastern Manitoba, Canada lies an inlet of the Lake of the Woods that is part of the United States. Also known as the Northwest Angle, this exclave of the United States, considered part of Minnesota, can only be reached from Minnesota by traveling over the Lake of Woods or by traveling through Manitoba or Ontario. More from Wikipedia.
2. Azerbaijan-ArmeniaBetween the Azerbaijan and Armenia border, there are a combined total of four exclaves or islands of territory that lie in the opposite country. The largest exclave is Azerbaijan's Naxcivan exclave, a not insignificant piece of territory located within Armenia. Three tiny exclaves also exist - two additional Azerbaijan exclaves in northeastern Armenia and one Armenian exclave in northwestern Azerbaijan. See this detailed map of Azerbaijan for the exclaves along the border.
3. United Arab Emirates-Saudi Arabia and United Arab Emirates-OmanThe boundary between the United Arab Emirates and its two neighboring countries, Oman and Saudi Arabia are not clear. The boundary with Saudi Arabia, defined in the 1970s, has not been publicly announced so cartographers and officials draw the line at their best estimate. The border with Oman is not defined. Nonetheless, these boundaries lie within a fairly inhospitable desert so boundary demarcation is not an urgent issue at this time. This detailed map shows the general area of the boundaries.
4. China-Pakistan-India (Kashmir)The Kashmir region where India, Pakistan, and China meet in the Karakoram Range is incredibly complex. This map illuminates some of the confusion but for a sense of the disputes, you can see Wikipedia's article on the topic.
5. Namibia's Caprivi StripNortheastern Namibia has a panhandle that extends far east several hundred miles and separating Botswana from Zambia. The Caprivi Strip provides Namibia access to the Zambezi River near the Victoria Falls. The Caprivi Strip is named for German Chancellor Leo von Caprivi, who made the panhandle part of German South-West Africa to provide Germany access to Africa's eastern coast.
6. India-Bangladesh-NepalLess than twenty miles (30 kilometers) separate Bangladesh from Nepal, "squeezing" India so that far eastern India is almost an exclave. Of course, prior to 1947, Bangladesh was part of British India and thus this border situation did not exist until the independence of Indian and Pakistan (Bangladesh was initially part of independent Pakistan).
7. BoliviaIn 1825, Bolivia gained independence and its territory included the Atacama and thus access to the Pacific Ocean. However, in its war with Peru against Chile in War of the Pacific (1879-83), Bolivia lost its oceanic access and unfortunately became a landlocked country.
8. Alaska-CanadaSoutheastern Alaska contains a peninsula of rocky and icy islands, known as the Alexander Archipelago, that cuts Canada's Yukon Territory as well as northern British Columbia off from the Pacific Ocean. This territory is Alaskan, and thus part of the United States, due to Russian colonization in this area in the nineteenth century.
9. Territorial Claims on AntarcticaSeven countries claim pie-shaped wedges of Antarctica. While no nation can modify their territorial clam nor can any nation act upon their claim, these straight boundaries that typically lead from 60 degrees south to the South Pole divide up the continent, overlapping in some instances but also leaving significant segments of the continent unclaimed (and unclaimable, according to the principles of the Antarctic Treaty of 1959). This detailed map shows the boundaries of the competing claims.
10. The GambiaThe Gambia lies entirely within Senegal. The river-shaped country was started when British merchants obtained the trading rights along the river. From those rights, The Gambia eventually became a colony and then an independent country.