The data question is huge for us. We have built our system on top of Hadoop + Spark in order to handle large amounts of data, and be able to apply computations to it. You're absolutely right that data is very heavy, and often the limiting factor, so we are doing everything we can to both get data into the system (currently with a focus on streaming data, since that gets around the heavy data problem), and getting algorithms to where the data is.
As far as specific algorithms, there are many that could apply. My personal experience is in machine learning algorithms, so I'm personally biased toward those. There are numerous algorithms like Classification, Clustering, Optimization, Anomaly Detection, etc. which are very CPU-intensive. We will be doing a blog post soon with a demo of live algorithms already in our system to give more concrete examples.
How is data for the algorithms provided? A lot of times it is big. And messy, and proprietary. There seems to be an implicit assumption that you can just plug different algorithms to different data sets. But I can't think of a project where that has been the case.
I also doubt that "algorithms" are the bottleneck in a lot of projects. I'm not an expert, but I have some personal experience to back up what people say about "data trumping algorithms" (e.g. Peter Norvig and others have written about this)
I would like to hear about some more concrete examples / success stories. "Algorithms" is just too vague. I think if this becomes successful, it will be by first narrowing it to a particular domain, and then generalizing it again.