Which PPN parameters are least well determined? There are by now (late 2010) many tests of GTR.  GTR is holding up quite well!  But, surely, we do not yet have it cornered in every possible way.  
In terms of the PPN parameters used to characterize low-order deviations of any theory of gravity from Newtionian ideals,  where are the loosest screws?   Which PPN parameter is least constrained by experiment?   What kinds of experiments would help constrain it?  
 A: From the articles linked in the comments:
Parameter  Bound        Effects                             Experiment
---------  -----        -------                             ----------
γ − 1      2.3 x 10^-5  Time delay, Light deflection        Cassini tracking
β − 1      2.3 x 10^-4  Nordtvedt effect, Perihelion shift  Nordtvedt effect
ξ          0.001        Earth tides                         Gravimeter data
α1         10^-4        Orbit polarization                  Lunar laser ranging
α2         4 x 10^-7    Spin precession                     Sun axis' alignment
                                                            with ecliptic
α3         4 x 10^-20   Self-acceleration                   Pulsar spin-down
                                                            statistics
ζ1         0.02         -                                   Combined PPN bounds
ζ2         4 x 10^-5    Binary pulsar acceleration          PSR 1913+16
ζ3         10^-8        Newton's 3rd law                    Lunar acceleration
ζ4         0.006        -                                   Kreuzer experiment

Where the parameters are so defined:

The parameters $\gamma$ and $\beta$ are the usual Eddington–Robertson–Schiff parameters used to describe the “classical” tests of GR, and are in some sense the most important; they are the only non-zero parameters in GR and scalar-tensor gravity. The parameter $\xi$ is non-zero in any theory of gravity that predicts preferred-location effects such as a galaxy-induced anisotropy in the local gravitational constant GL (also called “Whitehead” effects); $\alpha_1$, $\alpha_2$, $\alpha_3$ measure whether or not the theory predicts post-Newtonian preferred-frame effects; $\zeta_1$, $\zeta_2$, $\zeta_3$, $\zeta_4$, measure whether or not the theory predicts violations of global conservation laws for total momentum.

And these are the GR-predicted values:

γ = β = 1 and α1 = α2 = α3 = ζ1 = ζ2 = ζ3 = ζ4 = ξ = 0

