What problems involve large amounts of computation per byte of data?

Visualization:  Ray-tracing dynamic geometry with procedural surfaces could easily get up over 50 operations per input byte.  A compositing tree alone can easily get up to 50 operations per byte of input data.  A rib file, which holds all the data necessary to render a single image, can be fairly small but take hours to process. A lot of computer graphics for visual effects like fluid simulation, environmental effects like rain or snow, crowd simulation, etc, have a very small set of data, because you're only simulating expectation and not reality, and only to a minimum level of detail (i.e. semi-transparent white streaks on a black background could be enough detail to simulate rain.) Audio modeling and rendering.  (With a few bytes a midi file can require a huge amount of computation.) Regression analysis could pretty easily top 50 operations per byte of input data.

Non-parametric model discovery problems where you have to search through a large space of models for the one that best fits data. Bayesian network structure learning is one such problem: it is non-parametric and NP-Hard, so you have to do a lot of computation, even with a small number of variables and/or samples.