Unfortunately, the as of now accepted answer by @Sycorax, while detailed, is incorrect.
Actually, a prime example of regression through categorical cross-entropy -- Wavenet -- has been implemented in TensorFlow.
The principle is that you discretize your output space and then your model only predicts the respective bin; see Section 2.2 of the paper for an example in the sound modelling domain. So while technically the model performs classification, the eventual task solved is regression.
An obvious downside is, that you lose output resolution. However, this may not be a problem (at least I think that the Google's artificial assistant spoke a very humanly voice) or you can play around with some post-processing, e.g. interpolating between the most probable bin and it's two neighbours.
On the other hand, this approach makes the model much more powerful compared to the usual single-linear-unit output, i.e. allowing to express multi-modal predictions or to assess it's confidence. Note though that the latter can be naturally achieved by other means, e.g. by having an explicit (log)variance output as in Variational Autoencoders.
Anyway, this approach does not scale well to more-dimensional output, because then the size of the output layer grows exponentially, making it both computational and modelling issue..