Same issue here. I got an out of storage space message a day or two after updating to 8.4. I've always had gigs of free space on my iPhone. I delete some apps and other things and had 2gb free. A little later same message again, running out of space. I had only megabytes left of free space. So I go all out, delete videos, photos I don't want. Free space now showed 7.4 gigs. Next morning, 5.4 gigs free.
All of my previous free space plus at least about 4 gigs of additional space have been consumed by something since the 8.4 update. I had disabled the new music after having that on for a few days thinking it was casing this, but I'm still losing free space. I added maybe 20 songs to my available offline not even near close enough to account for the gigs of lost free space. I run the system as manual as possible. Nearly all apps are disabled for background updating and notifications. All notifications are off for apps. I try to disable all automatic updating, music downloads, app downloads etc.
Something is gobbling up gigs. I have even taken screenshots of the usage screen to see which category is growing, they aren't. The listed things such as photos, apps, video are not growing, only the shrinkage of my videos since I deleted them.