There's some hot ideas in here that are sure to upset Elhak and Dummy Thic some people. At least with the items, I'm pretty confident they're close to optimal. I might even write some articles on the math behind items in the future.
Also, please note that I have no idea if the positioning is correct. I just copied one of Soju's PBE comps and kept positioning the same way.
Dusk Riven
If you haven't heard, this is THE comp.
Stick some items and a GA on Riven, a Morello on Morgana, and do whatever the hell else you want. It's relatively impossible not to top two with this comp if you hit all your units.
If you manage to find a Chosen Dusk or Dusk Spat, take out either Janna or Zilean and put in Vayne for an easy life.
Ahri
Put items on Ahri. Buy an Aatrox. Watch Aatrox pull everyone into a ball. Ahri presses R. GG.
Similar to Riven, if you hit all your champions, it's a free top two.
Throw in pretty much anything at nine, it really doesn't matter. You should basically play around getting a big Chosen at nine, but again, the comp is giga free.
As a side note, the highest damage with two AP items is one Gauntlet and one IE. It's better than two Gauntlets, a Gauntlet and a Deathcap, or any other combination of two items. Expect an article on this soon, but just trust that I repeatedly did the math correctly for close to an hour.
Another side note, unless Ahri is taking damage, Shojin and Blue Buff do the exact same thing on Ahri. With either item, she'll need five autos to ult for the first time and six autos for every subsequent ult.
Definitely an Ahri Comp™
This is definitely an Ahri comp and not a Riven comp. Ahri has the damage items. It must be an Ahri comp!
Brawler Ashe
Probably the most popular one among my friends and the one I'm gonna get the most flak for.
Let's go over all the spicy choices I've made with this comp.
Firstly, we only have three Brawlers. The reason for that is quite simple: all the other Brawlers suck.
Secondly, Ashe seems to have some interesting items and seems to not have a Giant Slayer. Well, similar to my last choice, Giant Slayer sucks unless you're in the mirror. It's good into two-star Sett, Lee Sin and Sejuani, but overall, it's not going to do anything.
Now, something that is going to do something, is Shiv. Do you hate Riven? Dead. Do you have a ton of CC in your comp already? Swoosh. Do you have a broken ultimate with Spirit and two Bows? Brrr.
Note that this is the best third item over HoJ, Deathblade, RFC and Rageblade. I don't think there's possibly a better third item other than maybe GA.
Also, IE, Last Whisper is a crank.
Chosen Shade
Contrary to popular belief, you actually don't need Zed to be Chosen. You can run a Chosen Eve, but you should generally always roll for a level-three Zed regardless of who's Chosen.
Also, Spirit really helps the comp out. Throw in like a Sett, Lee Sin or Yone at nine if you can and you're good to go.
Enlightened Talon
Not much to say. Get some good items for Talon. Put in whatever champ you want at level nine. Top four?
¯_(ツ)_/¯
Chosen Sharpshooters
Fornari, why do you only run three Sharpshooters?
Well, because all the other Sharpshooters suck. Also, why the hell do people put Morello on Teemo? Put it on Jinx. Stop griefing.
Chosen Warlord
Go eight and roll baby. Also, if you get a bunch of Bows, you can stack Kalista as one of your carries.
However, the comp is pretty focused on three starring Katarina.
Also, I think Azir's soldiers should be directly to the left and right of Aatrox.
Chosen Diana
I have no idea if this is the way to run Chosen Diana. It's a hot idea that I do not stand behind.
Chosen Wukong
Hit level-three Wukong and get a bunch of frontline units. The comp isn't great, but it still should net you a top four.
I would recommending using the Wukong to just transition into Talon, but you do you fam.