By no damage, I mean compared to other AP damage dealing champions in their burst.
Ahri has no damage and I stand by that comment. Ahri's R is used for repositioning and safety. If Ahri is getting all 3 R charges off on you offensively, you're playing wrong. If she gets 2 of them on you offensively, she's playing well - so let's go with that. Her W likewise should be hitting minions instead of her target with the additional foxfires, if all 3 hit you you're playing it wrong.
So, with that in mind, let's take the base damage + scaling damage of the full combo of Ahri versus a few other AP champions at level 6 (I'm pretty sure it only goes downhill from there):
Ahri, LeBlanc, Lux, Annie, Malphite
First, think for a moment about what you expect out of this. Annie has low range and mobility, so she should clearly have the highest damage (but not by too much since she still has a shield and a great stun). Lux lacks mobility but has huge range to make up for that, then she has three utility abilities (snare/slow/shield) versus Ahri's two movement abilities so she should have less damage. Malphite is a tank and should have the least damage. LeBlanc has about as much mobility as Ahri and a hard CC ability so she should have almost identical damage.
I.e. Annie > Ahri = LeBlanc > Lux > Malphite
Ahri:
Base: 330
Magic Scaling: 185% AP
True: 90 + 35% AP
Totals: 420 + 210%
Annie: (assuming 1 enemy auto and 2 seconds for Tibbers AOE, i.e. the minimum if she hits a stun).
Base: 455
Magic: 305% AP
Lux: (assuming only 1 proc of her passive, which is proc'd not only by auto attacks but also her ultimate)
Base: 568
Magic: 225% AP
LeBlanc: (assuming she's maxing W for the waveclear instead of Q which would be even more damage)
Base: 475 if she misses the E, 515 if she hits it but loses range, 555 if she hits it fully
Magic: 260% AP if she misses E, 310% AP if she hits it but loses raneg, 360% if she hits it fully
Malphite:
Base: 430 + 15 per enhanced auto attack (not counting the auto attack damage itself since anyone could auto), so at 3 autos (easy thanks to slow and knockup of ult) that's 475
Magic: 210% AP
So what are the results?
Ahri's base damage is the lowest of the bunch:
Ahri < Annie < Malphite < LeBlanc < Lux
420 < 455 < 475 < 475 to 575 < 568
Ahri's scaling damage is basically lowest of the bunch:
Malphite ~= Ahri < Lux < Annie < LeBlanc
210% ~= 210% with 35% of that as True < 225% < 305% < 260% to 360%
So overall, all 4 champions have more damage than Ahri. Notice that I've made very fair assumptions for each champion's level 6 all-in combo.
Simply put, the only champion that Ahri can outscale damage-wise is Malphite and even that requires a large chunk of MR on her opponent after resistances thanks to Malphite's higher base damage.
Her base damages are pitifully low compared to Lux and LeBlanc, two champions that offer similar levels of utility (LeBlanc literally has the same amount of CC, probably better mobility with W and R, and a stealth passive in lieu of the MS boost, Lux has gigantic range alongside a snare, slow + vision, and a shield). Her scaling damage is only slightly lower than them, but pitifully low compared to an AP champion that has a bit less utility (although still potentially an instant AOE stun and a shield...) (Annie).
If you think my analysis is unfair choosing these champions (I do promise I only had an idea of the results beforehand, I didn't check every champion and come back with the best results. I picked champions I thought would be a fair comparison and did it as I typed) feel free to look up any other AP mid laner and compare the base and scaling damages and come back and let me know.
Ahri has no damage in her all-in compared to other AP mid laners. She may be safe in lane, but whoever she is playing against is safe in lane as well by default. She needs to poke you about 6-8 times with her Q on both directions before she has you low enough to all-in 1v1. If you're dying in lane 1v1 to an Ahri, you're not only playing wrong (as you've not dodged even 1 Q since she'd be OOM otherwise) but also playing too risky as you should have recalled after she poked you down to under 50% health.