Label propagation is a semi-supervised technique that makes use of the labeled and unlabeled data to learn about the unlabeled data. Quite often, data that will benefit from a classification algorithm is difficult to label. For example, labeling data might be very expensive, so only a subset is cost-effective to manually label. This said, there does seem to be slow, but growing, support for companies to hire taxonomists.
Another problem area is censored data. You can imagine a case where the frontier of time will affect your ability to gather labeled data. Say, for instance, you took measurements of patients and gave them an experimental drug. In some cases, you are able to measure the outcome of the drug, if it happens fast enough, but you might want to predict the outcome of the drugs that have a slower reaction time. The drug might cause a fatal reaction for some patients, and life-saving measures might need to be taken.