In Alaska? I wouldn't even presume to advise you what you'll need once you pull the trailer north on the AlCan highway... But since you're picking it up in the lower 48, here's my essential list.
tooth brushes and paste
SOME bottled water - some tap water in bottles
sooner or later you'll want to flush and your water tank will be empty - I once faced a crisis that could only be solved by flushing with "wine in a box" - and no - I didn't pass it through my kidneys first. A 2 liter bottle of ordinary tap water resides in my bath cabinet at all times. I use bottled water to cook with and to drink. Most campground water is safe, but a lot have high iron content and that makes horrid tasting tea. I've also boondocked and run out of water on the last day. Bottled water that's stored inside is tepid and can be used in a pinch as is for a sponge bath. Of course if you still have propane it's nicer warm.
canned soup-Progresso is my choice, not bad even cold - provides hydration and nutrition in a pinch
duct tape & Aluminum tape
lock tite & WD-40
bear repellant & wasp spray - both very effective on hostile humans too.
flares or flashers
first aid kit - replace every 2 years
needles and thread (I carry 2 sewing machines, but that's mega-overkill)
Waxed dental floss (good for a quick repair to an awning)
rawhide gloves (when you're trying to sew that effen awning by hand!)
paper maps - still a great adjunct to GPS
cell phone - get GREAT coverage, not cheapest
Plastic leveling blocks, yellow or orange - they weigh less than wood, but will break or wear out like wood
AirForums - people who will rescue you when the fertilizer impacts the ventilator!
Seriously - buy what you need when you need it or you'll buy way too many things that you use once every 5 years.
THOUGHTFULNESS - what can you buy that has more than one function? That should always be the best test. I have a large box end wrench that has been used to "adjust" the spare tire carrier up from a 15 inch to a 16 inch wheel, lift the weight bars on a Reese Dual Cam (if you have a bad back it's FAR better than the standard pipe because you push horizontally instead of lifting), drive tent stakes, kill a rabid small animal (no not my ex-husband), as well as tightening down big nuts (again, not the ex).