The symmetry dimension of a geometric structure is the dimension of its symmetry algebra. We investigate symmetries of almost quaternionic structures of quaternionic dimension <i>n</i>. The maximal possible symmetry is realized by the quaternionic projective space H<i>Pⁿ</i>, which is flat and has the symmetry algebra sl(<i>n</i> + 1, H) of dimension 4<i>n</i>² + ...