Typically, not much. Our system includes dynamic load-balancing and scalability so it can adapt quickly to dramatic changes in your traffic levels. (It is possible to create a site that's so inefficient that it can't keep up no matter what we do, but that's pretty rare.)
A major "slashdotting" (or insert the surge site of the week here) of a site hosted on our service will cost you (on average) less than $10, one time. There's no higher-tier pricing to get permanently pushed into, and we won't cancel you for having something to say that people actually want to hear.
This happens to one of our members about once a week, so you can bet we know how to handle it. Or rather, our systems do. Our load-adaptive clustering technology is at its best when handling demand surges, and our pricing is at its best when you'd prefer not to be billed based on a 1% event the other 99% of the time.
Note: Our service is based on science, not magic. It will not make slow sites run quickly, nor will it make sites scale if they aren't built for it. Your site must be able to complete requests at the same rate they come in. If it cannot, for whatever reason, then there is nothing our system (or any other) can do; you're gonna have a bad time. Please take this into account when designing sites intended for high traffic levels.