I have discovered that if you change the profile in JA Slideshow, save it, and change it back and save that, both the profile .ini files have permissions of 777. That is not good.
Just to be sure, I found that the version of JA Slideshow that is on the Joomlart site under the JA Biz template, which is 2.7.6, downloaded it and reinstalled it on the site, went and tested the profile change and yes, the version that JA has on its site has this behavior (so it's not like my version was hacked or something).
Please check this out and issue an updated version. I kept having random 777s pop up and finally realized that they were JA Slideshow profiles and that is what caused them.
Thanks for looking into this. I can give you my site information if you really need it; I have tested it on a couple of sites so far. You should get the same result.
Thanks!