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?

According to your definition infinite number of classes should be possible.
One way to think about a classification rule is as a division of feature space into segments. Hence, if we consider a one-dimensional feature space $x_1 \in \mathbb{R} $ as an input we can imagine a classifier that returns a class of "$k$" for any input within $[(k-1):k)$. So it would return "$1$" for $[0:1)$, "$2$" for $[1:2)$, etc.
Such a "model" is not estimated from any data but it is a classification rule.
But in practice having a classifier with an infinite number of samples will probably not be possible, because any dataset will be finite. However, we can imagine an "on-line" classifier that is re-adjusted as new samples come in. And in principle new samples can be of a yet-unseen class. Hence the number of classes can keep increasing forever and not be known in advance.
