We probe the merging channel of massive galaxies over the z = 0.3 - 1.3 redshift window by studying close pairs in a sample of 238 galaxies with stellar mass \lower 2.15 pt \hbox { $ \buildrel > \over { \sim } $ } 10 ^ { 11 } M _ { \odot } , from the SHARDS survey . SHARDS provides medium band photometry equivalent to low-resolution optical spectra ( R \sim 50 ) , allowing us to obtain extremely accurate photometric redshifts ( median | \Delta z| / ( 1 + z ) \sim 0.55 % ) and to improve the constraints on the age distribution of the stellar populations . Our dataset is volume-limited , probing merger progenitors with mass ratios 1:100 ( \mu \equiv M _ { sat } / M _ { cen } = 0.01 ) out to z=1.3 . A strong correlation is found between the age difference of host and companion galaxy and stellar mass ratio , from negligible age differences in major mergers to age differences \sim 4 Gyr for 1:100 minor mergers . However , this correlation is simply a reflection of the mass-age trend in the general population . The dominant contributor to the growth of massive galaxies corresponds to mass ratios \mu \lower 2.15 pt \hbox { $ \buildrel > \over { \sim } $ } 0.3 , followed by a decrease in the fractional mass growth rate linearly proportional to \log \mu , at least down to \mu \sim 0.01 , suggesting a decreasing role of mergers involving low-mass companions , especially if dynamical friction timescales are taken into account . A simple model results in an upper limit for the average mass growth rate of massive galaxies of ( \Delta M / M ) / \Delta t \sim 0.08 \pm 0.02 Gyr ^ { -1 } , over the z \buildrel < \over { \sim } 1 range , with a \sim 70 % fractional contribution from ( major ) mergers with \mu \lower 2.15 pt \hbox { $ \buildrel > \over { \sim } $ } 0.3 . The majority of the stellar mass contributed by mergers does not introduce significantly younger populations , in agreement with the small radial age gradients observed in present-day early-type galaxies .