  We study the power spectrum of dark matter density fluctuations in the
framework of the Effective Field Theory of Large Scale Structures (EFTofLSS) up
to three loop orders. In principle, several counter-terms may be needed to
handle the short-distance sensitivity in perturbation theory. However, we show
that a small number of extra coefficients are sufficient to match numerical
simulations with percent accuracy when a generic renormalization prescription
is implemented (allowing for running of the individual counter-terms). We show
that the level of accuracy increases with respect to the two loop results, up
to $k \simeq 0.4\, h$Mpc$^{-1}$ at redshift $z=0$, although the overall
improvement is somewhat marginal. At the same time, we argue there is evidence
that the behavior of the loop expansion in the EFTofLSS is typical of an
asymptotic series, already on the brink of its maximum predictive power (at
$z=0$). Hence, the inclusion of higher orders will likely deteriorate the
matching to data, even at moderate values of $k$. Part of the reason for this
behavior is due to large contributions to the (renormalized) power spectrum at
three loop order from mildly non-linear scales, even after the UV counter-terms
are included. In conclusion, the EFTofLSS to three loop orders provides the
best approximation to the (deterministic part of the) power spectrum in the
weakly non-linear regime at $z=0$, and higher loops are not expected to improve
our level of accuracy.
