Escapist Character: A king, partly divine, all but invincible, Super Strength, gets laid with every girl in his kingdom but snubs the goddess of R3D H0T S3XX0RS... Gilgamesh is wish-fulfillment Snark Bait by modern standards. He's effectively protected by a Grandfather Clause, since he predates most tropes.
Memetic Badass: The sole reason why the story survives today. We wouldn't have so many tablets if they weren't being made and put all over Mesopotamia just to tell everyone how badass he was and how you can never hope to reach the level of sheer badassery he was at.
Fridge Brilliance can be applied here when you figure that well, though he may have failed to achieve literal immortality within his own story; the Epic of Gilgamesh in the meta sense has become immortal and survives to this day.
Values Dissonance: The tendency to paint Shamhat as a harlot and seductress. Despite Ishtar's less-than-flattering portrayal in this story, she was a very important and revered goddess and the sex is part of the sacred duties of her priestesses. Not to mention that the weeklong tryst with Enkidu isn't a fall, but the first event that raises him above the level of an animal.
What an Idiot: If you have a plant that will make you immortal, don't wait until after you've taken a bath to use it.
He was actually going to give it to an old man to see if it would turn him young again, so he was probably playing it safe since he didn't know if it would actually kill him instead.
Woolseyism: Since some portions of the story have been lost, some translations feature original material to fill the gaps which occasionally works quite well. Most notably, at least one changes Gilgamesh's motivation from wanting immortality for himself to wanting to bring Enkidu back to life.