  We consider a completely resonant nonlinear Schr\"odinger equation on the
$d$-dimensional torus, for any $d\geq 1$, with polynomial nonlinearity of any
degree $2p+1$, $p\geq1$, which is gauge and translation invariant. We study the
behaviour of high Sobolev $H^{s}$-norms of solutions, $s\geq s_1+1 > d/2 + 2$,
whose initial datum $u_0\in H^{s}$ satisfies an appropriate smallness condition
on its low $H^{s_1}$ and $L^2$-norms respectively. We prove a polynomial upper
bound on the possible growth of the Sobolev norm $H^{s}$ over finite but long
time scale that is exponential in the regularity parameter $s_1$. As a
byproduct we get stability of the low $H^{s_1}$-norm over such time interval. A
key ingredient in the proof is the introduction of a suitable ``modified
energy" that provides an a priori upper bound on the growth. This is obtained
by combining para-differential techniques and suitable tame estimates.
