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.