Have you guys ever played Risk? It's a great game (pun totally intended). There is a board with a world map on it, each player gets a certain number of armies, and your goal is to take over the entire globe. This game was first invented in 1957, when the Cold War was really warming up. But even though Risk appeared after the height of global European competition over new colonies before World War I, it could still work as a great metaphor for that period of eager, active domination of other places on the part of the European states.
Indeed, Risk may be the best real-life game that we can think of to describe colonial competition among European nations, but the imperialists themselves were using the idea of a game to describe what they were doing long before we got here.
The inventor of the term the Great Game was a British intelligence officer named Arthur Conolly. He used this phrase to describe the struggle between Czarist Russia and Great Britain for control of India as a colony. This Great Game only ended in 1907, when Russia and Britain signed an agreement to end their struggles in favor of forming a united front against the increasingly powerful German Empire.
Kim is a player in the Great Game on the literal level, because his first real task for the Secret Service is stealing secrets from two agents of the Russian Empire in the Himalayas. But he is also a symbolic player in the Great Game, in the sense that he approaches the whole idea of imperial maintenance and British colonial power in India as a kind of game. Think about it: even his training exercises with Lurgan, where he learns to improve his memory and to imitate other people's speech more exactly, are basically just games of Memory and charades.
Kim is very serious about becoming a spy, but he plays around a lot to get there. Indeed, the Great Game, both as Kipling uses the phrase and in real life, makes colonization sound like fun—even though, of course, it's deadly serious and it costs innocent people their lives. But Kipling's whole portrayal of the Secret Service as a rich and exciting entertainment is probably the best recruitment pitch we can imagine for young British boys wanting to go abroad to play on the side of England while having a grand old time.