For chaotic dynamical systems, we use ideas from recurrence statistics and shrinking target problems to establish results on almost sure growth rates of extremes for dynamical systems. Along the way, we show how these almost sure growth rates can be used to estimate the local dimension of an invariant measure for a hyperbolic system (e.g. for the Hénon map, or the Lorenz equations). Work is joint with M. Nicol and A. Török.