Unlike many drives, the Apple Disk II has inputs that control the individual head stepper coils. These are commonly described as advancing by half-tracks, since advancing a full track requires activating two coils in sequence. Four-pole stepper motors, however, can not only be moved by driving coils individually, but also by driving pairs of coils. Since advancing by one pole moves a half track, energizing a pair of cells will access a quarter-track location between two half-track locations.
Experimentally, at least on the Apple //c, when writing data sequentially, it's possible to write data at 3/4-track intervals and then read back previously written tracks. For test purposes, I used a program that packs twelve double-high-res graphics pictures into the space of 27 normal tracks (2.25 tracks each) and allows random access via keyboard. I likely could have used 15 if I'd had that many DHGR pictures handy. This would likely not be reliable if tracks were being written by different drives at different times, and I'm not sure how reliable it would be for data written using one drive and read using another (I just have one drive on my Apple //c) but the fact that this works at all would suggest that it could be expected to be reliable if information were written on quality media using a drive specialized for the purpose--something that could have been really great for copy-protection purposes if a game would make use of data in a way that effectively required 200K to fit on a single disk ("normal" Apply formatting is 35 tracks of 16 sectors of 256 bytes, but the game Prince of Persia used a format that packed 4.5K bytes per track, which would allow 35 tracks to hold 157.5K); 200K would have exceeded that by a substantial margin.
Were there any other floppy drive systems that offered quarter-track control, or high-density systems that offered half-track control, or was such ability unique to the Disk-II style systems?