Is this E3 "predictions" or "hopes?" Seems a bit mixed.
Oh well.
Predictions
-Console upgrades, VR
-Electronic Arts shows off sports games, and we'll get our first proper look at Titanfall 2 and Andromeda (granted, they've been confirmed, so this isn't really a "prediction" per se)
-Bethesda will show off The Elder Scrolls 6, Prey 2 and something from Machine Games, if Prey 2 isn't that game already. Maybe a Wolfenstein sequel.
-Gears of War 4 stuff from Microsoft, likely a first proper look at ReCore, coupled with the confirmed appearances of Halo Wars 2 and Forza, and possible depiction of Rare's new game
-Nintendo spends most of their time with the NX, with The Legend of Zelda being the showcase games (granted, the latter is confirmed)
-Ubisoft will give us a first proper look at Watch Dogs 2, Wildlands, and an update on the status of For Honour
-Sega will showcase one or more Sonic games - Fire and Ice for the "Boomverse," and a new Sonic game for the mainstream series that's similar to Generations in that it will at least incorporate "Classic Sonic."
-Lots of stuff from Square Enix (just look at the Wikipedia article), but off said article, more info on Final Fantasy VII and another Tomb Raider game
-Resident Evil 7 is revealed (this is part wish, part prediction, but I'm going with the latter as I think there's a fair chance of it)
Wishes
I want a pony. Doesn't mean I expect hoofbeats. But fine, I'll make predictions that at least have a chance of coming true (i.e. no games from Nintendo, as it's been confirmed that Zelda is the only game they'll showcase)
-Halo 6 (I'm really not counting on this to appear at all, but hey, I can still hope)
-Bungie showcases the rumored Marathon remake (if revitalizing "old school" FPS is a thing now, then I'll gladly endure yet another Doom reboot if Marathon gets some love)
-Mistborn: Birthright is revealed, and is actually decent
-Perfect Dark is revealed in all its glory, in a sequel that pushes the series decades into the future so it doesn't feel out of place timeline wise, and it takes inspiration from Rucka's Perfect Dark novels and...hah hah, yeah, I can't do this.

-Resident Evil 7 is not only revealed, but is also everything that I could hope it to be (horror, but not a reboot)
-Activision/Blizzard reveals Metal Arms 2 (if only so I can get those characters in Heroes of the Storm)

-A Warhammer: Age of Sigmar game is revealed (while I have little love for the setting, I'm willing to give it a chance through game format, considering that in the wake of recent Warhammer FB/40K developments, there's certainly potential for good games within the IP)