In "The Day of the Doctor," the Moment shows the War Doctor his future, specifically Ten and Eleven. Why did she skip Nine? And if we're skipping around anyway, why not skip Ten, too? In other words, how did she decide which regenerations to include?
Out-of-universe, Christopher Eccleston was unwilling to appear, but I'm specifically looking for in-universe reasons.
In-universe, no reason is ever given.
However, if we had to speculate, we can imagine that it has something to do with the way that Ten and Eleven dealt with the Time War. The Moment seems to know everything about The Doctor's past and future incarnations, and it must have decided that Ten and Eleven were the best ones to pull back through time to help themselves make this decision.
Recall that, during Nine's short run, he was "very angry" about the Time War; in effect, he was running away from what he had done. Ten specifically calls himself out this when he leaves Rose in Peter's World with his meta-crisis self:
DOCTOR: Exactly. You were born in battle, full of blood and anger and revenge. Remind you of someone? That's me, when we first met. And you made me better. Now you can do the same for him.
By this point, Ten had largely come to grips with his actions during the Time War. We know that Ten knew, off the top of his head, how many innocent women and children had died on Gallifrey. This means he'd had time to reflect on his actions, and decide if they were justified on not.
Eleven, on the other hand, had the benefit of living through "The End of Time". (Based on Ten's comments at the start of that episode, regarding the Virgin Queen, it's likely that he was on his "farewell tour" when "Day of the Doctor" occurred.) He had been reminded, once again, of why he chose to do what he did, and what was at stake.
In other words, Ten and Eleven provided the War Doctor with perspective -- two slightly different perspectives, both with the benefit of time (decades or even centuries of time) to reflect on what had been done. Eleven even claims that he's thought a lot about what he would do differently, given the chance.
None of those things were likely to be true about Nine. His emotional state was too raw to think clearly; he hadn't had time to internalize his actions, see the ramifications, and judge if they were truly justified. He also hadn't had time to learn just how pointless they had been, given how many Dalek's survived. The Moment must not have felt he would have had the same effect on War Doctor that Ten and Eleven did.
On a side note: we can also justify why The Moment never didn't choose Twelve, or any future Doctor, instead. By the time Eleven regenerated into Twelve, he knew the Time Lords weren't dead, because Trenzalore had happened. So, anyone past Eleven wouldn't have been able to make the same decision with the same frame of mind as War/Ten/Eleven did. IMO, The Moment was less concerned with what decision they made, as much as how they made it.
Beside Mike's well thought answer, I'd like to add something, because I have also thought about that, specially after watching The Day of The Doctor for the 5th or 6th time:
The moment labels the future Doctors as "The man who regrets" (Ten) and "The man who forgets" (Eleven). Nine never actually got to regret his actions during the Time War (notice how he even gloats about it to the Dalek in Utah, in the "Dalek" episode: "I watched it happen. I MADE IT HAPPEN"), and of course he never really even pretended to forget about that.
Nine still had lots of anger towards the Time War. Ten started to make his peace about it, and Eleven was the first incarnation that really started to move on. The more I think about it, maybe Nine wouldn't have helped with the plan had he been present with his other three selves.
Now, I know what you're thinking: "But Nine still got together with the rest of the Doctors to save Gallifrey in the end", and yes, he did. But based on the footage we saw ("Now, for my next trick..." in The Parting of The Ways, his last episode), it was during the end of his life, right before he refused to go through with killing all the Daleks (and the humans on earth) again with the Delta Wave. He was presented with the same choice, and he didn't go through with it. That's when he started to change his feelings towars what had happened in the war. Also remember the Doctor forgets the events that take place when he meets with his future selves, due to "time streams being out of synch" or some timey-wimey thing that happens.
