  The growth index $\gamma$ is an interesting tool to assess the phenomenology
of dark energy (DE) models, in particular of those beyond general relativity
(GR). We investigate the possibility for DE models to allow for a constant
$\gamma$ during the entire matter and DE dominated stages. It is shown that if
DE is described by quintessence (a scalar field minimally coupled to gravity),
this behaviour of $\gamma$ is excluded either because it would require a
transition to a phantom behaviour at some finite moment of time, or, in the
case of tracking DE at the matter dominated stage, because the relative matter
density $\Omega_m$ appears to be too small. An infinite number of solutions,
with $\Omega_m$ and $\gamma$ both constant, are found with $w_{DE}=0$
corresponding to Einstein-de Sitter universes. For all modified gravity DE
models satisfying $G_{\rm eff}\ge G$, among them the $f(R)$ DE models suggested
in the literature, the condition to have a constant $w_{DE}$ is strongly
violated at the present epoch. In contrast, DE tracking dust-like matter deep
in the matter era, but with $\Omega_m <1$, requires $G_{\rm eff} > G$ and an
example is given using scalar-tensor gravity for a range of admissible values
of $\gamma$. For constant $w_{DE}$ inside GR, departure from a quasi-constant
value is limited until today. Even a large variation of $w_{DE}$ may not result
in a clear signature in the change of $\gamma$. The change however is
substantial in the future and the asymptotic value of $\gamma$ is found while
its slope with respect to $\Omega_m$ (and with respect to $z$) diverges and
tends to $-\infty$.
