Are there checks on Gilgamesh's power in the city? What are they, and how significant are they? Characterize the political organization of Uruk.
In short, no checks exist on Gilgamesh's power in Uruk. The political structure of the city is not democratic: Gilgamesh is the builder and supreme ruler of the city. He is described as "2/3s divine", and is more powerful than any other human. In fact, the driving plot force in the beginning of the epic is Gilgamesh's tyranny. He is abusing his power as ruler, taking brides on their wedding nights and over-working the men of the city. Thus, the people appeal to Anu, the god of the city, who in turn asks Aruru, the goddess of creation, to create a being who can rival Gilgamesh. This is of course Enkidu, whose personality and strength balances the king's. Thus, after his creation, Enkidu could be considered a check on Gilgamesh's power. He himself is quite significant in that he serves as the catalyst for Gilgamesh's quest for immortality.
As for the political organization of Uruk, Gilgamesh rules as a divine-right monarchy, in the most literal sense of the phrase. There are elders of the city, to whom he turns for guidance at certain points, but he is not bound by their suggestions. One may argue that he ultimately answers to the gods...but even that depends on which deity. He insults Ishtar regularly in the epic, defying her attempts to seduce him. It is clear that Gilgamesh is larger than life, denying his mortality and all challenges to his power.