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 ≳1011 M⊙, from the SHARDS (Survey for High-z Absorption Red and Dead Sources) survey. SHARDS provides medium-band photometry equivalent to low-resolution optical spectra (R ˜ 50), allowing us to obtain extremely accurate photometric redshifts (median |Δz|/(1 + z) ˜ 0.55 per cent) and to improve the constraints on the age distribution of the stellar populations. Our data set is volume limited, probing merger progenitors with mass ratios 1:100 (μ ≡ Msat/Mcen = 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 ˜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 μ ≳ 0.3, followed by a decrease in the fractional mass growth rate linearly proportional to log μ, at least down to μ ˜ 0.01, suggesting a decreasing role of mergers involving low-mass companions, especially if dynamical friction time-scales are taken into account. A simple model results in an upper limit for the average mass growth rate of massive galaxies of (ΔM/M)/Δt ˜ 0.08 ± 0.02 Gyr-1, over the z ≲ 1 range, with an ˜70 per cent fractional contribution from (major) mergers with μ ≳ 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.