I am wondering about the possibility of implementing a character leveling system that goes beyond a single game.

You could level your character, and in each game you would start with your character at its out of game level. For example, I play a game and complete it and my character levels to level 2 or 3. The next game I start, my character would start out at that level. You would level more by either fighting more opponents or setting THEIR levels higher. Could have variable skill levels and AI based on the level of the enemy rather than an overall AI difficulty. There could be rewards for defeating each particular set AI, or if you fight random enemy's the rewards could be randomized as well. The rewards could be something like: unlocking special buildings, unlocking technologies, unlocking shop items, starting items, special mounts. To balance this, it would be very important to give the options to give the AI special things as well. Maybe there could be a few ultimate AI's with some pretty crazy stuff as well as some amazing rewards.