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Annie, The Forgotten Child
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Greater Mark of Magic Penetration provides you with early game damage and compliments the high base damage of your spells
Greater Seal of Replenishment gives additional mana which helps to harass in lane
Greater Glyph of Ability Power increases the damage of your spells enabling easier last hitting with Q and increased burst damage
Greater Quintessence of Ability Power increased damage all around
Ignite - This helps greatly on Annie because you will often find that your burst is just shy of being able to kill your opponent. Although ignite does not do a massive amount of damage it often does enough to secure a kill that you otherwise would not have been able to get. Additionally this summoner spell becomes essential if you are up against an opponent like Vladimir or Swain who has a large amount of life steal. Ignite will reduce the health they are gaining back giving you an advantage when trying to duel them.
FIRST RECALL: you want to consider if your going to rush your Rod of Ages or your Rabadon's Deathcap. Rod of Ages will give you better lane sustain but less burst damage, and therefore it is a superior option if you are on equal or worse CS/kills to your lane opponent. It will allow you to survive if they choose to fight you, and its unique passive will heal you. Alternatively if you've gotten an early kill or two and more CS than your opponent you can rush your Rabadon's Deathcap which will give you more damage. The rational behind this is that if you are ahead of them in your build and levels you don't need to worry about sustain or having them kill you, and therefore you should focus on maximizing your damage which is after-all your main goal as a carry.
MID - LATE GAME ITEMS The order of items shown in the item list is more of a suggestion than a firm rule, you need to target your late game item order at how the enemy team is building. This section will outline what items you should focus on based upon the enemy team composition/build.
Rylai's Crystal Scepter - provides you with a good chunk of health which is useful if your being focused down in teamfights. Additionally if the enemy team's bruisers are focusing on you the slow from Rylai's will enable you to kite them.
Lich Bane - This greatly increases your burst damage against a single target and is best to get when you have a large amount of AP to convert into damage from Lich Bane's unique passive. The 7% movement speed doesnt hurt either for getting away from tanks or bruisers that may chase you.
Zhonya's Hourglass - This item gives a massive amount of AP but also provides an interesting way for you to protect yourself in teamfights. Beyond the armor increase the hourglass can make you invulnerable for a short time, enabling you to burst the enemy team, and then activate it when they turn to focus you. This is especially useful if you flash close to the enemy team to use your AOE stun, after the stun you often are targeted by the whole enemy team. Becoming invulnerable for a couple seconds can buy you the time you need to have your team protect you. Buy this quickly if you find yourself initiating fights with your Pyromania.
Will of the Ancients - This can give you the ability to regenerate your health over the course of a fight, but because Annie relies upon using all her burst at the start of the fight (when she is full health) it is not an ideal item for her. It can be useful if you have another AP carry (like Vladimir or Swain in top lane). It provides them with the aura and additional AP so it can be a valuable item for the group.
Usually you will start with your Disintegrate at level one to farm, but if your team is going to invade the enemy jungle (or if you suspect the enemy team will invade your jungle) you can start with Incinerate. You can use it a couple times at your base to get two stacks of Pyromania and then use it whenever its up until you get a complete stack of your passive. This will enable you to have an AOE stun at level 1, which is exceptionally powerful if you get into a team fight in the jungle. Once you've charged it up be aggressive about casting it. Many Annie players are passive because she is squishy, but at level 1 if you are able to flash into a group of their champions (who are likely guarding blue buff) and stun all of them you can get your team multiple early game kills.
You want to balance this ganking with lane pushing in order to keep your farm up. If you get a few kills but fall massively behind on your lane opponent you often will not end up with more gold than them. Balancing ganking and farming in lane is key to maximizing your gold production in the mid game.
Please leave comments, questions, or criticisms on this guide if you have any feedback. I read all of the comments on my guides and strive to improve them if something is missing. Thanks for taking the time to read it and have fun with this build AOEing down teams!