We have investigated scaling properties of the Aubry-Andr'e model and related one-dimensional quasiperiodic Hamiltonians near their localisation transitions. We find numerically that the scaling of characteristic energies near the ground state, usually captured by a single dynamical exponent, does not obey a power law relation. Instead, the scaling behaviour depends strongly on the correlation length in a manner governed by the continued fraction expansion of the irrational number β describing incommensurability in the system. This dependence is, however, found to be universal between a range of models sharing the same value of β. For the Aubry-Andr'e model, we explain this behaviour in terms of a discrete renormalisation group protocol which predicts rich critical behaviour. This result is complemented by studies of the expansion dynamics of a wave packet under the Aubry-Andr'e model at the critical point. Anomalous diffusion exponents are derived in terms of multifractal (R'enyi) dimensions of the critical spectrum; non-power-law universality similar to that found in ground state dynamics is observed between a range of critical tight-binding Hamiltonians.