These builds are, I'm just going to admit it outright, not great. Most of them are kind of stupid. But my god, most of these are hilarious, for reasons that will make a lot of sense. It's the LoL equivalent of seeing This Is The End. This is not something you get to try in solo queue, but if you're messing around with your friends, maybe in an AI game or 5v5 blinds? They are fun as hell, which is what you really need if you are done with a tiring day of solo queue.
AP Mid Soraka
Core Items: Rylai's Crystal Scepter, Liandry's Torment, Archangel's Staff
Not as goofy because this thing seriously works, especially with the passive AP shred from her Q synced up with Liandry's. In longer, drawn out fights, Soraka's heal can keep everyone alive, and the generous AP ratios mean both of her heals explode. She can make herself an extremely generous target, then use resistances and her heals to keep herself alive while punishing with Q spam and her silence.
There's also the fact that, comparatively, her silence lasts quite a long time, so it's useful for pinning down those champions with escape mechanics – her silence at max ranks is 2.5 seconds, whereas most stuns or roots last much less time, such as Sona's Crescendo. Land that on Talon and see him try to blow his escape now!
Core Items: Abyssal, Lichbane, Zhonya's
He's got a lot of magic damage based on his returns and the Thornmail he will eventually build, but the damage on it lands in that weird spot of magic damage on an AD champion, sorta like what Warwick has (though not as bad). AP doesn't help his Curl at all, because it's all based on returns...but it does help his Q out quite a bit.
You can always use the generous ratios on his ult to turn an entire area into a deathzone, and if you feel up to building something like Liandry's or Sorcerer's Shoes, it'll even sync up with the damage on Curl. But that's not why you do this incredibly hilarious build.
It is hilarious because the 1:1 ratio on his Q means he's dangerous not because he's going to pull you out of your team and do a ton of damage to you passively, like normal, it's because you're going to get smeared into the ground like when the Roadrunner passes over the Coyote.
Core: Wit's End, Lichbane, Nashor's Tooth
Not strictly AP, but Jax is interesting in that, much like a Master at Arms should, he's very good at building basically anything and coming out ahead with it. Sure, he does well with spellvamp and lifesteal because those items sync up with his ult's free resistances, but have you ever built just raw magic damage on him? A combination of new Nashor's and Wit's End makes it hilarious.
Since he's got AP ratios on his passive strike from his ult as well as his W, that slap hurts a lot, doubly so after he gets Lichbane and he's basically getting two AP ratios on that hit. And with the passives from the other two items I listed, not only are you doing passive on-hit magic damage, you're doing it fast, thus gaining more AP hits, and also stripping their resistances. How rad is that?
Core: Lichbane, Sorcerer's Shoes, Deathfire Grasp
You've seen this before on Phreak's champion spotlight, but most alternate modes from the Rift benefit well from AP Shaco. In the Rift, it's too important that he randomly appear in your lane and stick a knife in your back with that scary bonus damage. Instead, on Dominion and Howling Abyss – remember those? - building AP is just better for him, and a ton more fun.
His boxes do more surprise damage, and Shaco is too fragile to really get in close. He doesn't have the burst damage to be an assassin outside of jungling. Instead, get some AP and dance with your opponent. Use a combination of boxes and his Two-Shiv active nuke and DFG to dance on the outside, constantly moving back and forth and not closing the distance until they expose themselves. Lichbane will help you with that backstab damage if they ever get too greedy.'
Core: Lichbane, Zhonya's, Spirit Visage
AP GP is OP, man.
Core: Deathcap, Deathfire Grasp, Lichbane
This is a pretty stupid gimmick relying on two nasty factors – her dot, and her refreshing jump. Her ult has a pretty nasty ratio too, but it's hard to beat the kind of harass and safety from a mid laner AP burst mage with a long auto attack distance, and the harass damage from her last hitting's explosion. The fact that her dot has an AP ratio is silly enough to keep her going.
Remember that she's a complete inversion to normal Trist – normal Trist is a farmer with a few maybe decent tricks in the laning phase, while her base damages do enough to help out her early burst. AP Trist is an earlygame punisher who drops off even harder than most mages because her Q is effectively a dead skill. So it's not great, but it is hilarious literally jumping on someone to do hundreds of damage, blasting them into the wall, and hitting them with a 700hp Grevious Wounds, then jumping again.