It's slightly amusing to me that floppy disks were once called "HD", even though the abbreviation stood for high density rather than high definition.
https://en.wikipedia.org/wiki/2M_(DOS)
It worked by combining two completely separate techniques; the first, using more tracks, is pretty self-explanatory --- a lot of floppy drives, despite officially supporting 80 tracks, could actually move the head to 81st and 82nd tracks (rarely, even higher), and since the media was in the form of a uniform magnetisable disk, that let an extra 2 tracks worth of data be accessed.
The second technique was to reduce the number of sectors per track and the gaps between them, by making each sector bigger. To ensure interoperability with different drives rotating at different speeds, there's actually quite a lot of "padding" between each sector, as well as ID/CRC data for each one. If I remember correctly, in the highest capacity mode, 2M wrote only one huge sector per track instead of over a dozen in the usual format. This is similar to how hard disks have transitioned to using 4KB sectors internally, instead of the traditional 512B.
Everything is under the control of the drive, not the media.
They advertise their size as in: tell the drive what to expect. But the drive is in full control of where to write on the disk, and how densely.
As such, you could actually teach an old drive new tricks by coming up with a firmware that used a better encoding scheme, and then encoding the rest of your disk with that scheme.
† (though it ran on the CPU, rather than some microcontroller in the drive. "DOS" stands for Disk Operating System for a reason: a large part of it is a set of routines programs can call to make the CPU micromanage the read-head and stepping motor of the dumb disk-drive controller, with concepts like "files" being a mostly-optional thin wrapper above that, rather than some impenetrable abstraction layer.)
My first experience installing Linux was using a bunch of disks which were unable to handle even 1.44MB (even though they were sold as 2HD). I couldn't tell, of course, and I had no idea why my install kept on failing or generating errors part way through the installation process :(
Oh, that exists! From 1998: https://en.wikipedia.org/wiki/FlashPath
A floppy disk drive-compatible SD card adapter. Needs special software, though.
But in the end it's all about the density of the disk so I'm not sure if you could use a driver to store more data.
Also you will need very accurate positioning of the head. So the drive is also a limiting factor.
I think the largest floppies where 32MB.
http://www.amazon.com/Car-Cassette-Adapter-iPod-Sirius/dp/B0...
edit: Didn't realize they even made MemoryStick and SD Card floppy adapters. So this already existed, sort of! https://en.wikipedia.org/wiki/FlashPath
http://www.floppyemulator.com/
Note that i think you need to use special files that are read by the drive.
I think they are meant as a life extension for older CNC machines and industrial looms.
The second. With a better read/write head you could store more data on them (for example 2.88mb was available).
But the physical media of the disk may or may not have the necessary resolution.