Which country has won a gold medal in every Olympics?

Great Britain.

The fewest gold medals they won at a Summer Games is 1 at the 1904 Summer Games in St. Louis, 1952 Summer Games in Helsinki, and 1996 Summer Games in Atlanta. The most gold medals they won at a Summer Games is 56 at the 1908 Summer Games in London and 19 at the 2008 Summer Games in Beijing.
Great Britain