You might get a variety of answers to this question, however for me, you need look no further than Book 12 to see the lowest point of Odysseus in terms of his character. This is of course when Odysseus disregards the advice given to him by Circe regarding Scylla. Not only does he not inform his men of the danger that awaits them, he also is so consumed with his own arrogance that he believes he can fight Scylla and win. Of course, it is his men who pay the price, as Scylla snatches six men so quickly that Odysseus is unable even to land one blow on the monster. The way that the men scream futilely as they are butchered and the ship continues stands as a rebuke to Odysseus and his arrogance through disregarding the advice of Circe:
To me they cried aloud, calling upon me by name for that last time in anguish of heart. And as a fisher on a jutting rock, when he casts in his baits as a snare to the little fishes, with his long pole lets down into the sea the horn of an ox of the steading, and then as he catches a fish flings it writhing ashore, even so were they drawn writhing up towards the cliffs. Then at her doors she devoured them shrieking and stretching out their hands toward me in their awful death-struggle. Most piteous did mine eyes behold that thing of all that I bore while I explored the paths of the sea.
This, to me, is the lowest point of Odysseus, where he tries too hard to achieve super-human exploits. His over-confidence is shown by trying to do things the hard way and his loyal men pay the price. As someone who is charged with the safety and health of his followers, this represents a massive failure on the part of Odysseus.