  The symmetry dimension of a geometric structure is the dimension of its
symmetry algebra. We investigate symmetries of almost quaternionic structures
of quaternionic dimension $n$. The maximal possible symmetry is realized by the
quaternionic projective space $\mathbb{H}P^n$, which is flat and has the
symmetry algebra $\mathfrak{sl}(n+1,\mathbb{H})$ of dimension $4n^2+8n+3$. For
non-flat almost quaternionic manifolds we compute the next biggest (submaximal)
symmetry dimension. We show that it is equal to $4n^2-4n+9$ for $n>1$ (it is
equal to 8 for $n=1$). This is realized both by a quaternionic structure
(torsion--free) and by an almost quaternionic structure with vanishing
quaternionic Weyl curvature.
