I have been using a Moshi iVisor edge-to-edge for about 4 months and have had no problems with it. I have used it in conjunction with a UAG case and it fits perfectly. Note that it only has adhesive around the edges, and not on the screen itself. Others have reported the edges lifting and allowing dust in over time, but I have not yet experienced this (it is supposed to be fully washable and reusable, should this happen). Also it is one of those protectors with the 'grid of dots' on the glass (something to do with the way the glass is hardened, or touch sensitivity or something). This is only visible in direct sunlight when the screen is off, and I never really notice it.
http://www.mobilefun.co.uk/moshi-iv...ote-4-glass-screen-protector-black-p51561.htm
There are other edge-to-edge protectors, if you check out this thread (especially toward the end) you'll find others who have used them:
http://forum.xda-developers.com/note-4/accessories/tempered-glass-screen-protectors-t2878038
I'm thinking particularly of this one, which is similar to the Moshi:
http://www.amazon.co.uk/Tech-Armor-Samsung-Ballistic-Protector/dp/B00STN7SQY
Both edge-to-edge and those with a gap around have their pluses and minuses - but I, for one, can't stand the gap around the screen protector that the non-edge-to-edge protectors have, sorry but I think it looks dreadful. :/