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When I first started playing Anivia way back when, I used to always buy an early Tear of the Goddess and turn it into an Archangel's Staff right after my Rod of Ages in order to keep from having mana problems, typically followed by Zhonya's Ring both for more mana and to synergize with the staff.
Then they buffed Clarity and I started taking it. Not long after, the cost of Glacial Storm went down. Now Clarity alone gives me infinite mana early-game and is enough to sustain me through at least a couple of teamfights late game with only a Rod of Ages for straight mana. I no longer feel the need to build the Archangel's Staff and try to take the opportunity to build for more AP earlier in the game. But I've never been certain on exactly what items I should be buying after my boots and Rod of Ages.
I've considered a Void Staff, but a lot of the time my enemies don't stack magic resistance, and I don't know how little is too little for the 40% penetration to be worth it.
Abyssal Scepter is an alternative, but then you spend a lot for the magic resist, which I don't always need.
Lichbane is really good in theory. Anivia can hypothetically get more out of it than almost anyone except the likes of Karthus/Evelynn. Glacial Storm > attack > Frostbite > attack > cancel Glacial Storm > attack > Flash Frost > attack> detonate Flash Frost > attack > Frostbite > attack...but in real life, you won't be getting half of those procs (you usually have to Frostbite instantly after Glacial Storm to guarantee you get it off in time, trying to get both procs off of Flash Frost is moronic in a real fight, etc.), so I don't know if it really works as well as it sounds like it would, or it should wait until you have two other AP items.
I also just recently learned that the slow on Rylai's Scepter actually stacks multiplicatively with the slows on Glacial Storm and Frostbite, so that might be an interesting item to take. It kind of costs a lot for the AP, though...
Zhonya's might still be good, but probably not until at least your third AP item. I doubt it's cost efficient after just a Rod of Ages.
Deathfire Grasp might be okay for the cooldown reduction now that Glacial Storm is a 10-second cooldown. Sometimes I buy it if someone on the other team has a ton of health, but I forget to use it too often.
A lot of the time it seems like I just buy a couple of Blasting Rods as I can afford them and then sit on my gold until I can finish a whole item in one go or an obvious choice presents itself, which I'm sure is a bad habit. Does anyone have any advice on what to buy for the Anivia who doesn't need much mana/regen?