  In this paper boundary regularity for p-harmonic functions is studied with
respect to the Mazurkiewicz boundary and other compactifications. In
particular, the Kellogg property (which says that the set of irregular boundary
points has capacity zero) is obtained for a large class of compactifications,
but also two examples when it fails are given. This study is done for complete
metric spaces equipped with doubling measures supporting a p-Poincar\'e
inequality, but the results are new also in unweighted Euclidean spaces.
