Odin is highly associated with the number Nine. There are Nine worlds in Norse mythology, and He hung on Yggdrasil for Nine long nights. As I was typing this, I couldn’t help wondering if each of the Nine nights He hung there were meant to represent the Nine worlds.
It was during this Ordeal that He retrieved the Runes. The Runes are a method of communication and a method of divination. They are not Gods, but They are Their own spirits to be respected and worked with (as opposed to having Them work for you).
In order to retrieve something like that, even Odin had to sacrifice something–Himself. He had to give Himself over to Himself. In other words, He had to go from conscious, every day waking thought, to a deep meditation. Some people say He actually died on that tree (I agree with this), and that’s what allowed Him to travel to Helheim.
If, for a moment, we stop and consider that each of the Nine worlds could refer to different parts of ourselves, then it can be said that Odin took one day to travel through each part of Himself to reach a point–both physically and metaphysically–where He could access the Runes. Which could explain why the Ordeal lasted Nine nights–not longer or shorter. Nine.