Syndra
Even before her mid-scope update, Syndra was a powerful bot lane AP carry, due to being one of the best burst mages in League of Legends. Even before level 6, with the right support, she could easily threaten any ADC.
Due to her Dark Spheres, Syndra's DPS is not as sporadic as that of the usual burst mages, and it culminates with her ultimate that executes enemies below a certain health threshold!
Seraphine
Though Riot Games created her for the mid lane, and was relegated to the support role because of her control capabilities, Seraphine has been dominating the bot lane role by win rate.
This is due to the range of her basic abilities, which allows Seraphine to win every trade in lane against the enemy champions, especially since she has a heal from her W. And once Seraphine gets her Liandry's, with her low cooldown, she can spam her Q and E to put out consistent magical DPS from range!
Veigar
As one of the best scaling champions in League of Legends, it's a wonder why Veigar is not played in the bot lane more often, where he has a support to protect him during his weak early game and allows him to scale, just like the AD champions do.
During the late game, his Q and W are enough to one-shot any squishy enemy champion, and can be spammed, as the cooldown on his Q is low by default, and goes even lower with ability haste, while his W has a lower cooldown the more stacks of his passive Veigar has!
Corki
Corki is one of the forgotten League of Legends champions, as Riot Games seem to have given up on him due to how difficult he is to balance. This is because Corki deals hybrid, physical, and magic damage.
In addition to that his playstyle is far closer to that of an ADC than the previous entries on this list. And, unlike the other APCs, Corki has a dash ability that allows him to escape sticky situations.
Heimerdinger
Heimerdinger is one of the most versatile champions in League of Legends, who can be played in four different roles. One of those roles is, naturally, the bot lane role.
By positioning his turrets well, he can ensure that they always deal damage to the enemy, emulating the auto attacks of stereotypical ADCs.
And, for additional DPS, he can use his ultimate to summon a giant turret, that melts enemies better than any ADC can, due to how safe Heimerdinger is, by just leaving the turret in place and standing far away from it.
Ziggs
Ziggs is one of the best artillery mages in League of Legends, and as such, can abuse his large range to poke down any enemy bot lane. Something that is far harder to do in the mid-lane, which is full of assassins that can delete him or other ranged champions that can fight back.
In addition to that, playing in the side lane allows Ziggs to abuse the passive found on his Satchel Charge, which executes turrets below 35% of their health. This way, Ziggs' income increases, and he doesn't have to fight his mid laner for farm once the laning phase ends.
Karthus
Whatever role he's played, Karthus is one of the most dominant champions in League of Legends. However, he is so good in the bot lane role as an AP carry, that he is one of the best bot lane bullies as well.
This happens due to two things. The first one is his scaling, which makes KArthus such a dangerous champion overall. The second one is having a support to protect Karthus and lock down the enemies so that Karthus can unleash his damage on them even after death.
By doing this, the biggest counter to Karthus is avoided, namely the one where you try to burst him down and run away from his passive.
Cassiopeia
Battlemages in League of Legends are the AP champions meant to wade into the enemy team, and deal consistent damage. Cassiopeia is the best battlemage in League of Legends, due to her high skill ceiling.
This skill ceiling comes from her playstyle, where she needs to hit her Q to empower her auto attacks and deal a lot of damage to the enemy. With the low cooldown of her Q, and weaving in the auto attacks, Cassiopeia is the closest thing League of Legends has to an Ability Power carry marksman!