This is my first Build Guide thingy and in all honesty the only build I have made and finished in game. Anivia with this build I have destroyed numerous champions in 3-4 hits at lv 18. This build also surprisingly works for Dominion as well
Now all these items gear for ability power and in the end you should have the expensive items on the list but there is a few alternatives I have done and it still works decently. One is getting the MoonFlair SpellBlade instead of Lichbane if you want cheaper items. Another is getting Rylai's Crystal Scepter instead of the Void Staff if your dying alot. You lose Magic Penetration but you will live longer as it gives HP plus it gives more AP then the Void Staff. Now if the match goes on long enough or if you want to hurt much more with abilities switch out the Sorcerer's Shoes for a second Deathcap but do this late game when you can fight without having to run too much. This will make your AP over 900. Also you can take the Rod of Ages before the Deathcap if you are having trouble making money. If you are doing really well early game after you get the Sorcerer's Shoes and have like 1200+ gold before you go back to get the Archangel Staff, get the Needlessly Large Rod instead as it's ability power will help out alot.
These are mainly geared for AP but there are some Mana and Mana Regen ones as well. These can be interchanged in many ways but if I had to change em I would maybe get some more straight mana ones instead of a couple of AP ones or add a couple more Mana Regen's in but I like the way this is set up so I never change em. The attack speed runes are pretty much useless. I only have them because I was experimenting awhile ago and haven't got rid of em. I recommend you switch them out for either Mana Regen or more ability power, unless the amount of mana is bugging you a bit, then got for mana runes. Also straight mana or ability power runes can easily replace those gradual ones.
The reason that building the Skill Sequence that way is because if your with a teammate, preferably melee, cause an easy kill for your teammate or even yourself. Flash Frost stuns and as we all know stuns usually lead to kills if they are weak enough. For those who don't know how Flash Frost stuns, if the ball itself explodes on an enemy they get stunned, this can also be cut off pushing Q again which would make the ball explode sooner causing stunning at a closer range. Flash Frost also powers up Frostbite which, if I remember correctly, changes it's power to 350 at level 18.
The two spells are ideal cause flash is extremely helpful and ignite can help with getting kills. One change to these that I use more to be honest is Heal instead of Ignite because Anivia dies really quick if more then 1 enemy champion is after her at a time. You can also change either of em for ghost so you can get away easier if you need to. Take Revive instead of Ignite if you want to get back into the fight faster. I wouldn't recommend getting rid of Flash though as it can be used offensively as well. If you find you can't work with the Mana Regen you have then take Clarity instead of Ignite.
I use this sequence mainly cause I rarely use Crystallize so it doesn't help me much. This Skill Sequence has a big flaw however and that is you don't get the wall till late game. This can be both a blessing and a curse. For me it's a blessing cause I can't use the wall right till it's around level 3 anyway. You can easily change the skill sequence up and get the wall earlier. If you are soloing getting the wall earlier is essential! If this is the case take the wall at level 2 or 3 and max it before Frostbite as Flash Frost's stun is essential and you should always start with it regardless if you are soloing or not. Also if your paired up with either Ashe or Nunu you can get the wall earlier cause Flash Frost won't be needed to power up Frostbite since both those champions can also slow.
Pros / Cons
Now, like every build, There are pro's and con's to this build.
.Massive ability damage as this can get as high as 660ap without any changes to the items.
.Having a stun right off the start can lead to easy kills
.In late game forcing enemies to go through Glacial Storm by using Crystallize to nearly block a lane at max level can lead to good damage over time and multiple hits by Frostbite to get an easy kill.
.Flash Frost doubles as a way to escape because of the stun early game.
.Anivia is a decent solo as it is and this build just makes her deadly.
.Anivia is very squishy and since there is almost nothing towards armor or Mana Resist she can be easily killed.
.Even with this build she will still lose easy to Tryndamere or Teemo
.Anivia is slow and therefore can't run away easily without using flash or ghost and Flash Frost's stun doesn't last very long.
.Anivia is a decent solo but when she is alone she is also very susceptible to being ganked and she can't survive a gank very well.
.She is almost always targeted by multiple enemies.
.Anivia runs on Mana and Mana alone. Her physical damage is weak to be honest and since there isn't much for Mana Regen in this build you have to go back rather quickly sometimes.
.This build is very expensive. It is really hard to finish in 5v5 and Dominion but is near impossible to finish in a 3v3.