Dimensions of Hilbert Spaces confusion

If I understand it, Hilbert spaces can be finite (e.g., for spin of a particle), countably infinite (e.g., for a particle moving in space), or uncountably infinite (i.e., non-separable, e.g., QED). I am wondering about variations on this latter. The easiest uncountable to imagine is the cardinality of the continuum. What examples are there of other uncountable cardinalities in quantum physics (either more or, if assuming the negation of the continuum hypothesis, less)? Thanks.