I've always heard that Kan's $Ex^\infty$ functor has important theoretical applications, but the only one I know is to show that the Kan-Quillen model structure is right proper. What else is it useful for?
I'm looking for something more than the existence of a functorial fibrant replacement functor for the Kan-Quillen model structure, for which one can simply appeal to the small object argument.
As far as I know, the nice properties of the monad $Ex^\infty: sSet \to sSet$ are summed up in the following list (all model-categorical terms are with respect to the Kan-Quillen model structure on $sSet$):
$Ex^\infty$ preserves finite products.
In fact, $Ex^\infty$ preserves finite limits and filtered colimits.
$Ex^\infty$ preserves fibrations and acyclic fibrations.
$Ex^\infty$ preserves and reflects weak equivalences.
$X \to Ex^\infty X$ is an anodyne extension for every $X$.
$Ex^\infty X$ is fibrant for every $X$.
(Have I missed something?)
Thus, $Ex^\infty$ is a functorial fibrant replacement for the Kan-Quillen model structure which preserves finite limits, fibrations, and filtered colimits. The above list is rather redundant. For instance, I've listed (1) separately from (2) because I'm thinking of it as a "monoidal" property rather than an "exactness" property.
Some consequences are that:
The Kan-Quillen model structure is right proper.
The weak equivalences of the Kan-Quillen model structure are stable under filtered colimits.
The simplicial approximation theorem follows easily if you happen to independently know that $sSet$ is Quillen equivalent to $Top$.
For instance, I think I have the impression that (1) implies something important about simplicial categories, but I'm not sure what.
The applications I've listed can all be proven in different ways, and might even be inputs to proving the above properties of $Ex^\infty$. Ideally I'd like something meatier.