**Update to my above post, with full instructions and my current status, in case anyone else is in a similar bind (and in case this ends up working)**
I'm using an iPhone 5C (GSM). I got the "Slide to Upgrade" error while trying to go from iOS 7 (maybe 7.1.2?) to iOS 9 yesterday morning (09/21/15). Note that I do have a passcode set. I tried to update via wifi when I got stuck in this loop. I'm running the most current iTunes (12.3, 64-bit) on a fresh install to my basically new desktop running Windows 7.
My most recent backup was July 2014 to my iCloud (it's been full since then). My phone has never been backed up on any computer, so I can't back it up to a computer at this point (needs authority, which requires a working phone). I can't back it up to my iCloud account without buying more storage space (long story, but I can't get it to let me buy more- possibly a different issue, but ignore this for now), and even with more space, I'm not sure if I could back it up to my iCloud account.
Note that I can escape the screen by resetting the phone (hold both buttons until it resets), and then going through the motions again. This inevitably leads back to the "Slide to Upgrade" screen.
So, my goal is to escape the "Slide to Upgrade" error without losing data (as "restore" methods suggest), or to create a backup some other way before doing a "restore" method. This is my attempt at the former, following a combination of advice from the following two posts in this thread: Re: Ios9 stuck on slide to upgrade and Re: Ios9 stuck on slide to upgrade
1. Download iTunes (newest version), which I didn't have before this mess.
2. Download iOS 8.1.4 to my computer from https://ipsw.me/8.4.1 as suggested in this post from a few pages ago: Re: Ios9 stuck on slide to upgrade. (Note than I have an iPhone 5C GSM, and was previously running some version of iOS 7, so 8.4.1 would still be an upgrade relative to what I had two days ago.)
3. Connect my iPhone to my desktop, and reboot my iPhone into DFU mode (I did this from the "Hello" screen).
[To enter DFU mode, hold both the power and home buttons until the Apple logo appears (10 seconds), and then immediately release the power button but continue to hold the home button (about 3 more seconds).]
4. iTunes gives a message with the options "Cancel", "Restore", and "Upgrade". Hold "Apple" on a Mac or "Shift" on Windows when you click either "Restore" or "Upgrade" (I later saw someone say Alt for Windows; I used Shift)
4A. The first linked post (and many others) went the "Restore" route. I did not.
4B. I followed the second linked post and held "Shift" and clicked "Upgrade"
5. This asks you to locate the iOS file. Find the file you downloaded in Step 2, and open it here.
6. This *should* work....
However, I am currently stuck in another loop. At first, things seemed to go smoothly. It restarted my iPhone, and then the Apple logo appeared with an installation progress bar. After about 20-30 minutes, that finished. It then went black and immediately reset to just the Apple logo. It also disappeared from iTunes. My computer seemed to still be working on it (judging by the noises, although I was also working on other things), and after about 10 minutes the logo disappeared. After going to a blank screen for a second, it reset to the Apple logo again.
This has been repeating for about 2 hours now, although the computer no longer makes "busy" noises and the whole process now only takes a few minutes (steadily got fast for about an hour). Occasionally, the Apple logo will appear as a dark, metallic gray as it fades out, and sometimes the whole screen will be lightly overlayed with grayed vertical bars (about 8-10 across), kind of like a watermark on paper, as it fades out.
I have no idea what to do at this point. I suppose another reset is probably the best strategy. I'll update this if anything changes.