In 1867, the United States purchased Alaska from Russia, largely at the impetus of Secretary of State William Seward. It was a huge acquisition, but Seward wasn't done. He also considered buying Iceland and Greenland from Denmark.
Anna Andersen of the Reykjavík Grapevine tells the story. In 1868, Benjamin Mills Pierce, a mining engineer, compiled a report on the subject for the State Department. Greenland, he said, had impressive commercial fishing opportunities, as well as huge reserves of cryolite which were increasingly accessible with modern mining methods.
Iceland was less valuable, though its waters were rich with fish. The people were fiercely patriotic and might not take well to annexation.
But, Pierce argued, the greatest advantage to owning these territories would be strategic. If the US possessed Greenland and Iceland in addition to Alaska, it nearly surrounded Canada. This might help persuade it to join the United States.
The proposal never went anywhere. The Reykjavík Grapevine tells us that when Pierce's report was introduced to the Senate, the members literally laughed at the idea.