"Categorical" is not a well-defined mathematical term, so to answer this question we have to look to how this word is intended to be used. It is employed in contrast to "ordinal," "interval," and "ratio." One way to understand the primary distinctions is in terms of the groups of allowable re-expressions of the values. In the case of the latter three, there is an order that must preserved, whence all re-expressions must be monotonic (order preserving). For categorical variables, any bijection (including permutations) is ok:
Beyond that, anything goes with the nominal [categorical] scale.
(Stevens, quoted in the Wikipedia article.)
Another concept of "categorical" is that each outcome must be distinguishable from every other. This strongly suggests that any probability measure must be totally discrete: that is, all subsets are measurable, implying that each category will have its own well-defined probability. (This is not the case for continuous distributions.)
This would seem to indicate that the number of categories should be finite or at most countable, but that is not evident in the literature. For instance, an archetypal example of a categorical variable is a set of names. The set of all possible names on any finite alphabet is countable but not finite. It is therefore useful to allow countably infinite sets to be categorical. For example, if we are studying names given to babies, it is convenient to let the sample space consist of all possible names (rather than all names that we know of).
A slightly less realistic, but still conceivable, example of a categorical variable would be one that uses real numbers for names. In effect, such a variable would ignore all the usual mathematical structure on this set. I don't see any problem with such a usage, but it's worth observing that the axioms of probability imply that any probability distribution valid in this context would (a) assign a non-negative value to each real number and (b) would assign a non-zero value to at most a countable infinity of the reals.
One application involving an uncountable sample space that supports categorical random variables of infinite, even uncountable, support is the study of random graphs. To understand the rate of growth of some property of graphs, we would want to contemplate graphs on 0, 1, ..., $n$, ... nodes, so it's convenient to allow graphs to have countably many nodes. Random variables defined on this set can have various types. For instance, the mean vertex degree (if finite) could be considered of ratio type; the total vertex degree could be considered of ordinal type (and, therefore--by forgetting the ordering--is a nice example of a countable discrete variable). If we also allow a graph to have arbitrarily many edges and are interested in, say, its connected components, then we would have a naturally occurring category that is uncountable (because each connected component determines the subset of nodes it contains and there are uncountably many subsets of a countable set).
To summarize, it is reasonable to allow categorical values to attain an uncountable infinity of possible values, while recognizing that at most a countable number of them could ever have positive probabilities. This must be a discrete distribution, because all subsets are measurable, which is not the case for continuous distributions.