I've got 2 large boxes of floppy disks from the Atari ST and the Acorn Archimedes both of which I've been carefully preserving for decades in the knowledge that there is treasure therein yet unable to read them.
On the Atari ST especially, as it was possible to format a floppy with extra tracks and sectors to exceed the standard capacity, which is a feature I used a lot. Unfortunately this was rather dependent on the manufacturing tolerance of one's specific (SD or DD) floppy disk unit so there was no guarantee that another drive could read a disk formatted in this way.
At least, until recently-ish. I was a little late to discover it but there is a fantastic gizmo called a Greaseweazle which is a tiny PCB (bearing a chip or two) with a USB device port on one side and a floppy disk connector on the other. A standard 1.44Mb floppy has no problem with the quirky Atari formats.
It can read absolutely any format; it doesn't care. It'll rip a floppy using the raw flux (whatever that means) regardless of the number of sectors, what file system was used, or what device wrote it originally and save the image in a number of formats to your hard drive.
It has native support for Atari, Amiga, Archimedes, Akai, Ensonique, Sega, Commodore and PC disks but if you have floppies from an obscure bit of kit then it'll happily create a large (40Mb+) raw image of the magnetic properties of the floppy for later analysis. If nothing else it's good to know that whatever you can rip from a floppy can be preserved from further degradation.
More info here and here. I'm sure there are alternatives and other sites but I've just recovered some precious data that's been stashed on floppies in my boxes for decades and I'm, well, not unhappy about it
