Recovering data from factory-reset iPad I updated my iPad to iOS 8.3 and something or other failed, leading to a factory reset. I'd like to recover my files.

I found a couple Windows tools that apparently can do that, but either they can't install on the only Windows computer I have available at work (no admin privileges) or they're free trials that don't actually recover the files, just show that it can but want you to pay. 

I have not found anything promising so far, besides using PhotoRec per this link: http://www.cgsecurity.org/wiki/Recover_data_from_an_iPhone, which requires the iDevice to be jailbroken, which mine cannot be since it's 8.3 plus I want to keep overwriting to a minimum.

I'm running TahrPup, which is a Puppy "built from ubuntu 14.04 LTS trusty tahr packages" and can install from ubuntu's repositories/etc.
Number one, the more you use your iPad, the more chance that your missing data that might be recoverable will become overwritten so I have to mention that... it sounds like you already know.

Which generation of iPad is this? If it is an iPad 1, it should mostly be recoverable (either by a data recovery firm, or maybe some do-it-yourself recovery software such as Dr. Fone).

However, since you mention iOS 8.3, it must be a later generation iPad. Unfortunately for your situation, these are all hardware encrypted by default (whether or not you set a passcode), and once factory reset, to my knowledge it's impossible to decrypt the old data. Encryption like this is a security feature, but it also can kill the chances of data recovery like in your situation.

Your best bet would be to work from a backup of course (iTunes or iCloud), but I guess you don't have that option. I have heard that 3rd party app data (but not data from native apps) might be recoverable somehow but I'll have to check on that.

I know this isn't the answer you're looking for and I really hope you can scrounge up a backup from somewhere. We are seeing the always-on encryption of some devices now becoming a major problem for data recovery. It's better for security, but that necessarily also severely reduces chances for people to recover their lost data. We have seen some manufacturers backstep a little on this recently, but it seems like the overall trend is for more encryption.

You can still send this to a professional data recovery firm that can work on mobile devices as a last ditch effort, and if there is any chance, maybe if something slightly different from what we're discussing occurred, then they would have the best chance at recovering it. Most do not charge if they cannot recover the needed data.