Big Diomede Island is the easternmost point of Russia, in the Bering Strait. While it has no permanent population, it is host to a weather station and a military installation. Little Diomede, only a few km away, is the westernmost part of Alaska.
The island is known as Ostrov Ratmanova (о́стров Ратма́нова), Ratmanov's Island, in Russian. Actually located in the Western Hemisphere, the two islands are the only place in the world save for Antarctica where you can see land across the International Date Line and theoretically the easiest place in the world to cross between Asia and North America.
Getting legally in or out either from east or west is not that easy. There are no official border stations in either country. Almost everyone needs a visa for Russia, on top of this a special permit for the Chukotka region and the area being used by the military you will likely need another permit for that.
It is physically possible to reach th the USA from here, but there is no scheduled transport, and Little Diomede has no customs or immigration facilities.