Are there classifiers with infinite number of classes? I am writing something about time series classification and wrote the following:

A classifier is a function that for given input $x_1, ..., x_n$ (in this case representing data stored in audio file) returns $y \in C = \{c_1, ..., c_k\}$, where $C$ is a finite set of classes (like $\{M, F\}$).

But then I realised that it might not be true. However, I can't think of any example of a classifier with an infinite possible number of classes. Do they exist? If so, what are some examples?

The Distance dependent Chinese restaurant process may be an example. You can think of it as a clustering method, in which the number of clusters is adaptively chosen based on the data. As you add more data points, the number of clusters can in principle increase without bound. So there are an infinite possible number of clusters (or classes), but for any finite amount of data, only finitely many of the clusters are occupied.
