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We present the first constraints on the full parameter space of the Galileon modified gravity model, considering both the cosmological parameters and the coefficients which specify the additional terms in the Lagrangian due to the Galileon field, which we call the Galileon parameters. We use the latest cosmic microwave background measurements, along with distance measurements from supernovae and baryonic acoustic oscillations, performing a Monte Carlo Markov Chain exploration of the nine-dimensional parameter space. The integrated Sachs-Wolfe signal can be very different in Galileon models compared to standard gravity, making it essential to use the full cosmic microwave background data rather than the cosmic microwave background distance priors. We demonstrate that meaningful constraints are only possible in the Galileon parameter space after taking advantage of a scaling degeneracy. We find that the Galileon model can fit the Wilkinson microwave anisotropy probe 9-year results better than the standard ?-cold dark matter model, but gives a slightly worse fit overall once lower redshift distance measurements are included. The best-fitting cosmological parameters (e.g., matter density, scalar spectral index, fluctuation amplitude) can differ by more than 2σ in the Galileon model compared with ACDM. We highlight other potential constraints of the Galileon model using galaxy clustering and weak lensing measurements. [ABSTRACT FROM AUTHOR]