Methods that find insightful low-dimensional projections are essential to effectively explore high-dimensional data. Principal Component Analysis is used pervasively to find low-dimensional projections, not only because it is straightforward to use, but it is also often effective, because the variance in data is often dominated by relevant structure. However, even if the projections highlight real structure in the data, not all structure is interesting to every user. If a user is already aware of, or not interested in the dominant structure, Principal Component Analysis is less effective for finding interesting components. We introduce a new method called Subjectively Interesting Component Analysis (SICA), designed to find data projections that are subjectively interesting, i.e, projections that truly surprise the end-user. It is rooted in information theory and employs an explicit model of a user's prior expectations about the data. The corresponding optimization problem is a simple eigenvalue problem, and the result is a trade-off between explained variance and novelty. We present five case studies on synthetic data, images, time-series, and spatial data, to illustrate how SICA enables users to find (subjectively) interesting projections.