I'm using Capitally (https://mycapitally.com) which I started building to replace my 3 spreadsheets that needed constant attention and broken down quite often. It now completely covers tracking my investments - spreadsheet nr 1, still two to go - portfolio strategy and budgeting.
There is an app for iOS - seene.co. The amount of movements you have to do to capture enough pixels is prohibitive for my taste. I think that google has nailed it - it's super simple.
As for storing the layers - you would only have the "from above" pixels, and only a few. Probably that's why there is only a LensBlur in their app in the first place.
If you just want a small displacement effect like on depthy - then the key is no sharp edges in the depthmap. We will try to tackle this in one of the upcoming updates...
As for the fill, if the depthmap is of good quality and without sharp edges - there is no need for that, unless you go berzerk with the scale of displacement.
It's exactly that - single image, and a depth map calculated from a series of shots made upwards. Both these are bundled in a fake-bokeh image. There is obviously no more pixels. However...
If you choose your subject wisely, the effect can be pretty believeable, using a simple displacement map.
On iOS though it's something different. There's no depth map in there, just a flat image moving counterwise to your hand movements.
I've created a parallax viewer for lens blur photos. It's an open source web app available at http://depthy.stamina.pl/ . It lets you extract the depthmap, works on chrome with webgl and looks pretty awesome on some photos.
There is quite a few things you can do with this kind of images, so feel free to play around with the source code on github https://github.com/panrafal/depthy