iOS does have a file system, perfectly usable by any app that uses Document Pickers - Office, Pages/Numbers/Keynote, Pixelmator, MindNode, etc.
I actually think the bigger strength of Document Pickers is that it enables automatic integration with Dropbox, SFTP apps (Transmit) and other tools for getting files moved on and off my iPad. For the most part, I don't try to carry too much stuff with me, but with SFTP, I have secure access to my NAS from anywhere with WiFi, and I keep copies of files synced as needed via other ways if I need a local copy. It's just a shame that not more apps actually enable access to it currently, or hide it behind unfamiliar controls (Cabinet, I'm looking at you).
But for productivity, these tend to be where I spend my time:
- Things & Fantastical (Time Management)
- Office (Excel and Word mostly)
- Paper (Design / Investigation Notes)
- Cabinet & Pocket (Research Articles and PDFs)
- Transmit, Dropbox, Documents (File Storage / Access)
Hobby productivity:
- Ulysses (Writing / Blogging)
- Lightroom / Pixelmator (Early Tweaks on Photography)
- Web Browser / SkySafari (Control of my Astrophotography Rig, using a Web-based Linux Stick)
But it's still worth pointing out that this really shows that the iPad is part of my productivity, rather than the whole package. I don't write code on it for my job. And I do head over to the Mac to do heavier processing of photography. Writing is about the closest I can get to fully doing on an iPad, since Ulysses can export to Word for final tweaks and saving as a PDF. But I like it since I can continue work on both the iPad and Mac.
But it handles everything that I want to do away from my desk right now well enough, and is the lightest device I have for doing so. The fact that I can research, take notes on that research, and mull over my design decisions for code from the iPad is honestly plenty. Trying to do complicated coding tasks is something I've gotten far too used to having big screens for, and I also like the big screens for heavy photo editing.