I'm on iPhone 5 under Windows 10 Pro (I know this forum is iTunes Mac but I don't see a comparable thread under iTunes for Windows, and it looks like this issue applies to both Mac and PC).
My iOS version is 8.3, so I upgraded iTunes to the latest 12.3.044 on the computer in anticipation of upgrading phone to 9.0.1.
The first thing I noticed in iTunes 12 was the Sync button was grayed out and the Backup button would not do anything.
I clicked File > Devices > Transfer Purchases from 'iPhone' and that seemed to work in a reasonable amount of time.
Then I navigated to iPhone > Apps > File Sharing and was able to xfer files from various apps to the computer.
Next I tried rebooting Windows and the iPhone (holding down the power button and home button for a full reset).
Navigates to iPhone Settings > Summary > Backups section and this time the [Back Up Now] button works.
Undern section iPhone, I clicked [Update] to go to iOS 9.0.1 and it said I had not transferred purchases.
I had done this already before the reboot, but just to be sure, I again clicked File > Devices > Transfer Purchases from 'iPhone'.
Again it worked but this time MUCH more slowly! It took upwards of 25 minutes where a few minutes earlier it had taken 5.
For a while it looked like Transfer Purchases was locked up, so I hopped onto Google which led me here :-)
That said, the Transfer Purchases seems to be done, but after reading this thread I am questioning whether I should skip updating iOS to 9.0.1.
Any advice?