Yes, Annie carry.

Let me explain.

Annie's Q, Disintegrate; has a 4 second cooldown, does 80% AP and refunds the mana cost and half the cooldown on kill.

This means when farming, Annie's Q has 2 second cooldown and almost uses no mana.
Her using no mana for her Q means she can regenerate mana more effectively than someone like Ezreal can. Ezreal currently gets mana from playing aggressive in lane, Annie gains it by playing passive.

And the damage on her Q goes from 80/115/150/185/220 magic damage, which is pretty high.

Now let's add in runes.

When finally getting to champions, Aery is the best for DPS Annie at the moment.
Why?
Aery adds on 10% AP scaling to abilities.
This changes Annie's 80% AP Q into a 90% AP Q.
And it can proc from auto attacks too, allowing for Annie to poke with autos if she's using Q to stack up her Pyromania.

The last thing to add is Annie's E, Molten Shield.
This has a 10 second cooldown, which is pretty generous for her whole 3 seconds of 16/22/28/34/40% damage reduction. Yes, damage reduction. And her damage reflection that comes from it, 20/30/40/50/60 magic damage reflected every time someone auto attacks you, +20% AP scaling.
You can utilize this best when you have a support who shields or heals a good amount, so you can Q and E them and have them think they can outdamage you.

Also, Tibbers is an amazing meatshield. You could even say Tibbers acts as a spawnable support. You want to always have Tibbers in front of you when you summon him, as he can soak skillshots.