My biggest gripe about the Civ series is that it felt empty after a while. I love hearing the stories
other players experience when playing, but there was never a good way to recreate and share those experiences.
What Civ needs is challenges and leaderboards. Pre-configure scenarios that everyone can play and compare their experiences together. Score each play on a number of criteria (cities built, fastest tech, army size, etc), and maybe even have custom achievements for that particular scenario - like how fast you fully explored the particular valley the scenario starts you in, or how quickly you defeated the aggressive city state that starts next to you.
Not only should randomly generated maps be shareable, but Sid Meier should have weekly scenarios that involve hand crafted challenges. Not just normal games with the AI trying to act like a human, but the AI existing as part of the map. Example: at turn 50, an aggressive hoard comes down off the mountain attacks your civ. When you reach the other side of the mountain, there will be a Civ with 4 cities and an advanced army. You have certain diplomatic options available (like a quest to fulfill) or you can try and challenge them. How you handle these various challenges will impact future scenarios.
The important thing is that the AI isn't trying to play the same game. It's just there as part of the map to give the player an interesting experience.