The Epic of Gilgamesh is written in the spirit of the great sandbox adventures Skyrim and Fallout. Your character is a demigod, living in a world populated by mortals and other demigods. This world in influenced by non-corporal deities whose actions and motives are seemingly chaotic, but otherwise act as a playing field leveler.

Your goal as a player is to engage in epic adventures that are worth writing down. Throughout the game, your achievements are recorded by scribes for posterity. The more notorious your deeds, the more likely they will be copied and disseminated through history.

What are those deeds?

Military Conquest

Your player can raise an army and overthrow the governments in other cities. If you manage to take over the world, history may remember you. At least until the next great conqueror comes along.

Monster Battles

Your player can do battle with demons and monsters sent from the Gods. The more powerful the monster, the more fame you receive. It may be posthumous fame... but that's literature for you.

Sexual Prowess

There are several figures in history (male and female) who are famous by virtue of sleeping their way to the top. Or bottom. Or something truly kinky...

Why should the heros get all the glory? There were plenty of harlots too.

Competitive Sport

Heros sometimes found fame by dominating in competitive sports. Be them races, or games of chance, or duels, or ritual combat. History remembers those that excel.

Mystical Journeys

Heros regularly and often set out on mystical journeys. Some even made it back. And while they often fail at what they set out to achieve, when they do make it back they return with insights and wisdom. Or at least some scars and stories to tell at the bar.

Die with the most toys

Finally, players can try for fame by being filthy rich. Build a financial empire. Construct palaces. Have a legion of trophy spouses. Sure, you can't take it with you. But the mound they bury you in might be legendary in its own respects.

Make the trains run on time

Sometimes a hero is remembered for taming waves, not making them. Running your chunk of civilization competently, judiciously, and keeping your people safe from outside invasion is a feat in itself. Sure, Hammurabi conquered plenty of stuff in his time. But we remember him for making sense of the legal system.

Renaissance (wo)|man

Some heros were not particularly great at any one area listed so far. But their successes and failures in MANY areas made them great. There will be a track to immortality that covers half-assing in enough fields to make a total ass out of yourself.
Each of the ways of gaining fame will build up a player's prestige in a different way. Levels of prestige will bestow boons, and the maximum level of each prestige track bestows literary Immortality. And for fun, you can tune into what the scribes are writing about you throughout the game.