A lot of people seem to have to wrong idea about how programs use memory. A game like Civ4 should be EXPECTED to use more memory as the game progresses. This is because many of the things that you see in the game (e.g. units) are probably represented in the running program by one or more "objects", each of which uses memory. In the early part of the game, there are few units, cities, tile improvements, etc. As the game progresses, the map becomes more complex, cities grow and multiply, and there are lots more units. The memory to represent all those new objects is dynamically allocated from the operating system as it is required. So, the memory used by the program will naturally rise as a game progresses.
That being said, Civ4 uses a _lot_ of memory for what it does. It's kind of shocking to see a game require about a gigabyte of RAM to run efficiently, but I suppose we'll look back 10 years from now and laugh at how games used to run in "just" 1 GB.