  The extended Jordan-Brans-Dicke (eJBD) theory of gravity is constrained by a
host of astrophysical and cosmological observations spanning a wide range of
scales. The current cosmological constraints on the first post-Newtonian
parameter in these simplest eJBD models in which the recent acceleration of the
Universe is connected with the variation of the effective gravitational
strength are consistent, but approximately two order of magnitude larger than
the time-delay test within the Solar System. We forecast the capabilities of
future galaxy surveys in combination with current and future CMB anisotropies
measurements to further constrain the simplest dark energy models within eJBD
theory of gravity. By considering two cases of a monomial potential (a quartic
potential or a cosmological constant), we show how Euclid-like galaxy
clustering and weak lensing data in combination with BOSS and future CMB
observations have the potential to reach constraints on the first
post-Newtonian parameter $\gamma_{\rm PN}$ comparable to those from the Solar
System.
