  Quasiregular mappings $f:\Omega\subset\R^{n}\to \R^{n}$ are a natural
generalization of analytic functions from complex analysis and provide a theory
which is rich with new phenomena. In this paper we extend a well-known result
of A.~Chang and D.~Marshall on exponential integrability of analytic functions
in the disk, to the case of quasiregular mappings defined in the unit ball of
$\R^n$. To this end, we first establish an ``egg-yolk'' principle for such
maps, which extends a recent result of the first author. Our work leaves open
an interesting problem regarding $n$-harmonic functions.
