  We report on a direct search for elastic photon-photon scattering using x-ray
and $\gamma$ photons from a laser-plasma based experiment. A gamma photon beam
produced by a laser wakefield accelerator provided a broadband gamma spectrum
extending to above $E_\gamma = 200$ MeV. These were collided with a dense x-ray
field produced by the emission from a laser heated germanium foil at $E_x
\approx 1.4$ keV, corresponding to an invariant mass of $\sqrt{s} = 1.22 \pm
0.22$ MeV. In these asymmetric collisions elastic scattering removes one x-ray
and one high-energy $\gamma$ photon and outputs two lower energy $\gamma$
photons. No changes in the $\gamma$ photon spectrum were observed as a result
of the collisions allowing us to place a 95% upper bound on the cross section
of $1.5 \times 10^{15}\,\mu$b. Although far from the QED prediction, this
represents the lowest upper limit obtained so far for $\sqrt{s} \lesssim 1$
MeV.
