I think you're right, it does have the possibility to be a really good fit for wireless Arduino-based devices. DigiKey has the chip for $12.51 (http://www.digikey.com/product-detail/en/CC3000MODR/296-35512-1-ND/3768439), which is a lot better than official wireless board or even the WiFly adapter from SparkFun et al.
I've got the data sheet bookmarked and will hopefully get to reading it this weekend, but I glanced through it at lunch and it seems doable. The chip's 3.3V only so you'll have to level convert the SPI pins but there's several ways to do that.
You're right it doesn't have a built in webserver for HTTP, but I don't think it would be too difficult to adapt the existing Arduino libraries to do that.
You've probably seen this guy's working on a shield (http://www.evenchick.com/cc3000-shield-and-the-internet-of-things.html) but I'm not sure of the project's current status. TI's got several eval boards, I'm going to look into those as well.
Please keep us posted on any progress you make, and I'll do the same.