  Gray & Hervella gave a classification of almost Hermitian structures (g,I)
into 16 classes. We systematically study the interaction between these classes
when one has an almost hyper-Hermitian structure (g,I,J,K). In general
dimension we find at most 167 different almost hyper-Hermitian structures. In
particular, we obtain a number of relations that give hyperK\"aher or locally
conformal hyperK\"ahler structures, thus generalising a result of Hitchin. We
also study the types of almost quaternion-Hermitian geometries that arise and
tabulate the results.
