Dominions and Dominions II are meant to be playable by e-mail. This means that you don't have to be on your computer except when you actually play your turn, and that with many players they wouldn't even play one turn at the same time. Of course, then it might take a day or two before one turn is finished... but after stuff starts happening, it helps that you are forced to think things through.
Dominions 1 (Priests, Prophets and Pretenders) is still being sold, but even if you can't get the full game try out the demo. You can play any of the 14 nations, although you can't recruit the few of the more powerful commanders or special units. 14 nations is still quite a lot, and hwile human nations might't have that different units, nations like Abysia, Caelum or Jotunheim are VERY different from other nations. Also, unlike the DomII demo, there is no time limit... You can spend
hundreds of turns playing the demo alone. The user interface is pretty different from most other games, especially in Dom1.
If you haven't quessed it yet, I own both DomI and II, and am a big fan of the developers (two Swedes that made the games as a hobby!).
If buying a game is still no-no, search PBEM from Google. PBEM means Play-By-E-Mail, and these games can't be that different in how they handle battles. There is also some other term that defines this system, something like "simultaneous turn generation", which means that every command from every player is handled at the same time, instead of Player 1 doing everything before Player 2. If any kind of battle is present in a game like this, it can't be directly controlled by player.
Some quick Google hits:
General
http://newgame.agava.ru/eng/a_general.html
From wikipedia, , the term seems to be "simultanously-executed".
Diplomacy, which is both a board game and a (new ?) computer game.
http://en.wikipedia.org/wiki/Diplomacy_%28game%29
And the demos of the games of Illwinter. I'd be happy to answer any questions about these games, as I'm rather addicted:
Dominions: Priests, Prophets and Pretenders (Dom 1)
http://www.illwinter.com/dom/downloads.html
Their first game, Conquest of Elysium, with annoying no-save restriction and Civ-like graphics:
http://www.illwinter.com/coe/downloads.html
Dominions II from the publisher's homepage:
http://www.shrapnelgames.com/illwinter/d2/6.htm