Restricting the attention to the case of fixed parameters support, it's my understanding that (minimal) sufficient statistics of fixed dimensionality, i.e. a fixed number of of them, exists in, and only in, the case of exponential families.
Looking outside, I found the case of the Cauchy distribution:
where "reduction" is a achieved by the sorting, which is not invertible. I don't find this example too interesting as order statistics seems to be always sufficient for iid data and the dimensionality is equal to the dataset size.
I also found the following (very) related question:
but I fail to understand how the first example doesn't have fixed dimensionality of 1, while the second example is too loosely defined.
What is a known example of a distribution whose minimal sufficient statistics dimensionality isn't fixed nor equal to the dataset size (ideally sublinear)?