jats:titleAbstract</jats:title> jats:pStationary compact astrophysical objects such as black holes and neutron stars behave as classical systems from the gravitational point of view. Their (observable) curvature is everywhere ‘small’. Here we investigate whether mergers of such objects, or other strongly dynamical spacetimes such as collapsing configurations, may probe the strong-curvature regime of general relativity. Our results indicate that dynamical black hole spacetimes always result in a modest increase jats:inline-formula jats:tex-math</jats:tex-math> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> mml:mrow mml:mo∼</mml:mo> </mml:mrow> mml:mn3</mml:mn> </mml:math> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="cqgacb9cdieqn1.gif" xlink:type="simple" /> </jats:inline-formula> in the Kretschmann scalar, relative to the stationary state. In contrast, we find that the Kretschmann scalar can dynamically increase by orders of magnitude, during the gravitational collapse of scalar fields, and that the (normalized) peak curvature does jats:italicnot</jats:italic> correspond to that of the critical solution. Nevertheless, without fine tuning of initial data, this increase lies far below that needed to render quantum-gravity corrections important.</jats:p>