  Using controlled numerical N-body experiments, we show how, in the collapse
dynamics of an initially cold and uniform distribution of particles with a
generic asymmetric shape, finite $N$ fluctuations and perturbations induced by
the anisotropic gravitational field compete to determine the physical
properties of the asymptotic quasi-stationary state. When finite $N$
fluctuations dominate the dynamics, the particle energy distribution changes
greatly and the final density profile {decays outside its core} as $r^{-4}$
with an $N$-dependent amplitude. On the other hand, in the limit where the
anisotropic perturbations dominate, the collapse is softer and the density
profile shows a decay as $r^{-3}$, as is typical of halos in cosmological
simulations. However, even in this limit, convergence with $N$ of the
macroscopic properties of the virialized system, such as the particle energy
distributions, the bound mass, and the density profile, is very slow and not
clearly established, including for our largest simulations (with $N \sim
10^6$). Our results illustrate the challenges of accurately simulating the
first collapsing structures in standard-type cosmological models
