It's gone from 2.8 gigs to 3.43 gigs in just a week on my Lumia 635, will using a storage cleaner app make it purge temp files? I change my start background several times a day every day. Is that why it keeps growing?
I doubt that backgrounds have anything to do with that. From my observations the system folder grows with OS/firmware updates. It also seems to contain some certain cached files that are somehow crucial for the system, thus aren't not stored in the usual temporary folder, where WP's very own storage sense could easily remove them. Any so called Cache Cleane apps from the store won't help you there either. These apps basically don't do anything besides filling your entire storage, then prompt you to reboot your device to let WP's very own cache clean mechanisms kick in.
try flushing your browser (IE) cache / Cortana search cache
Then what apps get heavier usage ?
Have you checked STORAGE SENSE?
Have you deleted TEMPORARY files? (hint, it reduces SYSTEM ever so slightly)