Dominions 3 is a fantasy strategy game from Illwinter Game Design, published by Shrapnel Games. It's expensive and has lousy graphics and an extremely steep learning curve, but once you get into it it's one of the most complex and rewarding experiences you'll find. The game features over 1500 units spread over dozens of nations, more than 600 spells and 300 magic items that can be combined in a breathtaking variety of strategies to crush your opponents.
The game can be played against the computer, but it truly shines when played online against other players. Here each participant connects to a server and submits orders for the next turn; typically there will be a new turn every 24 to 48 hours, and a long game can continue for months.