My iOS 9 upgrade story...Ipad Air 2.
I upgraded several phones and 1 other iPad in in our family with no problems. When I upgraded my iPad Air 2, after the upgrade I got the black "connect to iTunes" screen. When I did, it said it was in recovery mode and needed to upgrade or restore. I selected both (many different times). Nothing happened. I would just get stuck on the same screen and I would do a hard reset.
Then I found a thread where you had to hold the home key down when connecting to iTunes to go into device recovery mode. I did that and was able to get it to go through the whole process of re-installing the iOS 9 upgrade. When this process finished, it stayed stuck on the white screen with the Apple Logo. I had to do a hard reset (hold home and power buttons) to get it to quit. I did this several times also. I even left it overnight, thinking it might take longer. Nope.
Next, I read on the internet how to downgrade to 8.4.1. So, I did that, and it worked!!! I was able to restore a backup and I was back in business. But, since I can never leave well enough alone, I tried the upgrade to iOS 9 again with the same results as above.
Then, I decided to try one more thing. I downgraded to 8.4.1 again, but this time I chose "set up as new iPad" rather than restoring my old backup. Then, I told it to upgrade to iOS 9 over the air (not from iTunes). Voila! That worked great, no problems. The drawback is that I now have a factory-reset iPad Air 2 running iOS 9. This is ok for me because I really don't keep anything important on it, so I am happy to start over and redownload the apps I need.
It seems that there was something in my configuration of apps, etc. on my iPad that prevented the initial upgrade process from completing. But, there was nothing wrong with the iPad itself. So, starting with a factory reset iOS 8.4.1 allowed me to upgrade with no problem to iOS 9.
I hope this helps other people. Good luck.