I've found a way to keep myself interested in the game. I thought I'd share it, since it really works for me and I'm not that odd, at least relative to most civ III players.
What is critical is to be truly invested in the outcome, so the process is actually exciting. What is proving to work for me is keeping a winning streak alive.
1) Chose a level where you are very competitive -- for me, that's monarch.
2) Play all random, so you get wild map formations and new civs to increase variety
3) Play out every start with no cheating.
4) See how many games you can "win" in a row.
Personally, I define a win as retiring with a multiple tech lead and by far the largest civ. No need to play out foregone conclusions.
Each fresh start has taken on a materially heightened sense of importance since a bad position could bring the streak to a halt. For example, last game I drew Rome and was placed on a rotten piece of turf, round and barely bigger than a full size city. That connected via a 4-tile isthmus to a lush paradise of green tiles with water, extra shields, lux everywhere, and Bismarck. It looked like a long shot, but a dense pack build in Rome's lousy starting spot permitted a vet archer rush that caught Bismarck shorthanded in the capital. There followed an interesting little war where Berlin's now abandoned limited road system acted as a base for our defense -- the roads turning archers into essentially the equivalent of horses. Germany still had more cities and attacked in force, only to get mowed down. After a palace bounce, about 90% of production occurred in our new German holdings. But the point is that it was fun precisely because the start presented a challenge to the streak.
During later stages, I now tend to be less cavalier and to think much more seriously about the game. If you want a long streak of wins, you can't take life-and-death chances without near certainty, or the law of averages is going to bite you sooner than you expect. Are oil and rubber going to be available? Concern with outcomes is what makes the game fun.
It hasn't happened yet, but one of these games is going to see the AI gang up successfully and may involve trying to play back into contention after an unsuccessful ancient era. That too, I bet, will be fun to try.
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