Anu is the sky-god in Mesopotamian mythology and the doting daddy of Ishtar.
Being the sky-god and all, Anu probably has a lot on his plate. So, he doesn't have a very big role in The Epic of Gilgamesh—but he does turn up at several key junctures
The first time this happens is in Tablet 1. When the citizens of Uruk are outraged at Gilgamesh's outrageous behavior towards the young men and women of the city, Anu is the guy who hears their prayers for help. And he takes swift action to set things right by getting the goddess Aruru to make another man (Enkidu) who will be equal to Gilgamesh, and thus able to confront him on his own terms. Weird, though, that he delegates this task to a goddess, right?
We see Anu again in Tablet 6, when Ishtar flies up to heaven in a huff because Gilgamesh has rejected her offer of marriage. When Ishtar asks to borrow the Bull of Heaven, Anu resists at first, but he eventually gives in. Anu is clearly a devoted daddy who can't help spoiling his children a bit. That said, Anu isn't afraid to put his foot down when he thinks it's called for.
In Enkidu's dream at the beginning of Tablet 7, Anu is the god who says that either Gilgamesh or Enkidu must die for killing Humbaba, cutting down the most majestic tree in the Cedar Forest, and killing the Bull of Heaven. This is rather odd behavior, really.
First of all, it seems the killing of Humbaba is what got Miss Ishtar (Anu's daughter) so hot for Gilgamesh to begin with. Then, it sure seems like Anu doesn't think that taking the Bull of Heaven to earth just to teach Gilgamesh a lesson is such a great idea. So, then, why does he decide that for killing Humbaba (clearly a fame-seeking endeavor) and the Bull of Heaven (clearly self-defense) one of them must die? And, of all things the determining factor is whoever cut down the biggest tree? (Wasn't Enlil the one that cared so much about the Cedar Forest? Is Anu sucking up?)
At any rate, Anu puts his foot down about one of them dying, and, well, we all know how that turns out. To us, it seems like Anu is more of a distant arbiter of justice than a god who really cares about creation, like Shamash. (And he probably has a thing or two in common with other sky gods.)