What you need to bring is really up to you. You would be able to show up with nothing and be just find, but I wouldn't recommend it. I would recommend, at the very least, bringing extra socks, underwear, one bar of soap, and one deodorant. Yeah, you'll be able to get stuff from the C-store but you may not always be able to go there. Beast is full of things that take time and going to the C-store is something that isn't necessarily going to happen when you need it (IE if you run out of soap it could be a few days before you have time to go to the C-store).
On that note, I would recommend bringing about $100 in cash. The cadre have more free time than the New Cadets because while you guys are getting briefed and doing tests, we'll be mostly free. Chances are that more often than not your squad leader will go to the C-store during one of those periods where you all are being briefed on something and he/she is just waiting for you guys to finish. It's simpler if you can just tell them what you need, give them cash, and then have them go pick it up for you.
Lastly, unless you receive mail stating that you need to show up in a t-shirt, shorts, and low quarters I recommend arriving in civilian clothes. Casual civilian clothes that you would wear regularly (so long as they're not inappropriate) along with regular shoes. I recommend this because unless you're dressed like a prostitute, no one is going to judge you on what you wear when you show up. One of the first things you do after you leave your parents is change into issued pt shorts, low quarters, and an issued white t-shirt. And that's what you'll wear from that point until you change into White over Gray later. What you actually wear to R-day will probably go in a bag that will most likely get locked up with all your stuff that you're not allowed to have (phone, etc etc etc). Chances are at some point as a plebe you'll be somewhere that you'll wish you had civilian clothes for, and unless you're going to have them mailed once the AY starts, it's nice to have already.
I'm slightly confused about the CBT Packing List posted on the admissions portal. How much of this 'recommended' stuff is actually not covered by West Point? For instance, it recommends bringing a razor and shaving cream and a toothbrush and toothpaste, none of which are mentioned on the 'issued items' section. Are things things not provided? If not, that seems to contradict being able to show up with nothing and be fine.
Lots of things are issued but not all of it is sufficient. For instance, I remember running out of the issued deodorant pretty early in the game. Also, you're not issued an abundant amount of soap so it's nice to have some when the issued stuff runs out (you only get issued a small bar).
It's been said many, many times before but remember you have to carry everything you brought plus what you are issued for a 12 hour day. Our DS last year hung the boots over his shoulder, wore the low quarters and had the running shoes plus all of the other gear - multitool, parachute chord, $100 allowed cash, bible, recommended copy of paperwork, 6 inch fan, sock liners, compression shorts.... all stuffed into one of those tiny sports back packs. The kind of backpack that has strings for straps. On R Day, we couldn't believe the size of some of the duffle bags others had in line and were bringing with them. When combined with what they were issued - there was no way they could carry it all. We did mail a package with more gear after the first phone call. Figure out what works best for you and press forward as Tugboat says.
You do have to carry everything, but not for long. There's a point before you start going through the process where Cadre screen what the NCs have. At that point, if they have anything they're not supposed to or if they brought huge duffle bags those will be taken and locked away until the end of Beast. They'll get some time to retrieve whatever they need from that bag as well. So you won't necessarily have to carry everything you bring with you.