AWS can't exactly do this for you, as explained by previous commenters, but they do give you the basic features you'd need for implementing it.
AWS Costs & Usage Analytics http://www.newvem.com/new-relic/
This is 1st hand information from a AWS Rep I once dated.
I'd assume since it's a spike in costs, Amazon has more interest in keeping you as a customer, at your much lower typical usage, than to have you abandon, even if you have a huge bill.
Obviously if this was a typical problem, you would need to rethink your architecture and infrastructure needs.
In either case, expending a few neural cycles prior to the action would be beneficial.
I'm all for being able to initiate billing inquiries online--in fact, I'm ditching Bank of America because I can't dispute charges on line.
But in Amazon's case, this isn't a dispute, rather this is asking them to forgive a (hopefully) one time transgression, which is much easier to do by speaking to someone, than filling out a form.
I find it similar to haggling for a temporary reprieve in my fantastically high cable bill.
I think thought that it's a failure of technologists to properly address the challenges of billing.
For example, what if you have set up a bunch of auto-scaling instances to handle heavy load. During the middle of the night, your application suddenly goes viral in Indonesia or Brazil or some other large country in a different timezone from yours.
Your bill goes over its threshold - what does Amazon do? Do they terminate some of your instances (potentially losing production data!)?
The problem is that the exact times that "intelligent" bill-limiting features are likely to be used are the exact instances in which they could have absolutely disastrous impact if they make a mistake[0]
I haven't experienced this myself, but I've heard Amazon is actually rather lenient when it comes to waiving fees for clear mistakes (accidentally provisioning an expensive instance that you didn't mean to) - their customer support is supposed to be very good, though I have not had to use this myself.
[0] By "mistake", I mean that they are following your instructions correctly, but your instructions happen to be incorrect for handling this situation.
It's not terribly difficult to implement one's own bill limiting system using the API.
The right way to do it (currently) is to model the costs yourself and tie it in with your operational automation, then determine what your best course of action would be during a shitstorm. Generally, I would wager the best option is to replace your front page with a static html "overwhelming response" page, maybe with some light vector graphics.
If you have opted in for the "don't let me spend more than $x," then yes.
Create tiers of things that email / SMS additional people as the numbers go up if you want to be very careful etc...
Amazon voided all charges without question. Their level of customer support is off the charts. But AWS would never be my first pick to start up something. Your costs will be way way lower when going dedicated. Obviously you should be able to scale, but I think startups generally overestimate their success when it comes to that, and wasting a lot of money for things they have no use for yet.
You could have your AWS deployment strategies ready for when the need would arise..
(Having said that, AWS could really do a lot better with it's notification targets than just email.)