Abstract: Magnetic fields are expected to play a key role in the dynamics and the ejection mechanisms of the hypermassive neutron stars produced in binary neutron star mergers. The understanding of the magnetic field amplification, triggered mainly by the Kelvin-Helmholtz instability during the merger, is still a numerically unresolved problem due to the relevant small scales involved. Here, we perform high resolutions, large-eddy simulations to capture these small-scale dynamos produced during the merger, following the newly formed remnant for up to several tens of milliseconds. For the first time, we find numerical convergence on the magnetic field amplification in the remnant, where the average intensity saturates at ~10^{16} G around 5 ms after the merger. We find neither clear hints for magneto-rotational instabilities, nor significant impact of the magnetic field on the redistribution of angular momentum in the remnant in our simulations, probably due to the very turbulent and dynamical topology of the magnetic field. The comparison between simulations with different initial magnetic configurations show that the initial topology is quickly forgotten in a short timescale of few milliseconds after the merger. Moreover, at the end of the simulations, the average intensity and the spectral distribution of magnetic energy over spatial scales barely depend on the initial configuration.