What are some examples of Odysseus being boastful or showing an infatuation of self in Homer's Odyssey?
At the end of the Trojan War, Odysseus declares that he himself won the ultimate battle. He declares that he alone thought of the the idea of creating a Trojan Horse. Odysseus is filled with excessive pride. He refuses to acknowledge the gods as having any direct connection with his human affairs, even though Poseidon sent a sea monster to destroy the Trojan who did not believe the Trojan Horse was a gift.
Truly, Odysseus does not acknowledge the gods along his journey. He is self sufficient and filled with pride. He does not need help from the gods. He is a strong, self-willed man. He uses his wit and strength to make it through each obstacle along his way while trying to reach Ithaca:
In the Odyssey, however, we are given an opportunity to see Odysseus at the center of the stage, doing what he does best—getting out of difficult situations as easily as he seems to get into them—but in a much different light.
Odysseus uses his wit, guile and intelligence to overcome each obstacle found in his path. He does not need the gods because he is self sufficient. He is smart. He understands how to overcome each obstacle. He refuses to acknowledge the gods as resources.
Proudly, Odysseus shouts out his name to the Cyclops as he is leaving his territory. While it would have been better to keep his identity unknown, Odysseus proudly yells out his name to further torment the Cyclops. Odysseus and his excessive pride cause more problems for Odysseus and his men. Now Poseidon, the father of the Cyclops, is angry and determined to destroy Odysseus.
Truly, it is clear that Odysseus is being tested by the gods. He has one struggle after another struggle while trying to reach Ithaca. For years, Odysseus encounters major problems while trying to reach home.
Some argue that Odysseus is all about himself. He does have so much pride until he endangers himself as well as the lives of his men:
Some have argued that too much of the "human" side of Odysseus shines through, that he is nothing more than a grasping, greedy, selfish, disreputable man who simply bides his time, does as little as possible to help anyone else, and always makes sure he takes care of Number One first.
Ultimately, Odysseus has no choice but to cry out to the gods. Stranded on a raft in a stormy sea, Odysseus cries out to the gods. He finally admits that he needs help. He acknowledges the gods by crying out in a frustrated tone of voice, asking what do the gods want from him. In this moment, Odysseus admits that he cannot make it on his own. He needs the gods to finish his journey. Athena comes to his rescue.
If we look strictly at the text of the Odyssey, we find Odysseus moving back and forth between responsible and irresponsible words and deeds. As mentioned by lsumner, perhaps the key moment in Homer's Odyssey comes in Book 9. After Odysseus has managed to escape from the Cyclops' cave, he reveals to him his real name (earlier he had told the Cyclops that his name was "Nobody"):
So they argued, but could not daunt my ardent spirit, and I shouted to him again in anger: “Cyclops, if any man asks how you came by your blindness, say that Odysseus, sacker of cities, Laertes' son, a native of Ithaca, maimed you.” (A.S. Kline translation)
Although Odysseus' cunning and trickery had managed to help him and his men escape from the Cyclops' cave, Odysseus, earlier in Book 9, had ignored his men's pleas to leave the Cyclops' cave before the monster returned because HE wanted to "see the giant himself, and test his hospitality".
We also find some suggestion in Odyssey 10 that Odysseus stays with Circe for an entire year out of self interest. Eventually, his crew has to persuade him to leave. Thus, Homer writes that "My proud heart yielded to their words." (A.S. Kline translation)
Whereas Odysseus' revelation of his name to the Cyclops results in Poseidon's wrath against him, in Odyssey 12, Odysseus' own self-interest leads him to have his men stuff their ears with wax and bind him to the mast of the ship so that he can hear the Sirens' song.
Of course, at the conclusion of Odyssey 21, we hear Odysseus utter the ultimate boast, but in this instance, who can blame him? He is about to slaughter the suitors. It is a boast James Bond himself could be proud of.
"The guest in your hall has not disgraced you. I have not missed the target, nor did it take me long to string the bow. My strength is undiminished, not lessened as the Suitors’ taunts implied." (A.S. Kline translation)