We have constructed merger trees for galaxies in the Illustris simulation by directly tracking the baryonic content of subhaloes. These merger trees are used to calculate the galaxy-galaxy merger rate as a function of descendant stellar mass, progenitor stellar mass ratio, and redshift. We demonstrate that the most appropriate definition for the mass ratio of a galaxy-galaxy merger consists in taking both progenitor masses at the time when the secondary progenitor reaches its maximum stellar mass. Additionally, we avoid effects from 'orphaned' galaxies by allowing some objects to 'skip' a snapshot when finding a descendant, and by only considering mergers which show a well-defined 'infall' moment. Adopting these definitions, we obtain well-converged predictions for the galaxy-galaxy merger rate with the following main features, which are qualitatively similar to the halo-halo merger rate except for the last one: a strong correlation with redshift that evolves as ~(1 + z)^2.4-2.8, a power law with respect to mass ratio, and an increasing dependence on descendant stellar mass, which steepens significantly for descendant stellar masses greater than ~2 × 10¹¹M⊙. These trends are consistent with observational constraints for medium-sized galaxies (M* ≳ 10¹⁰M⊙), but in tension with some recent observations of the close pair fraction for massive galaxies (M* ≳ 10¹¹M⊙), which report a nearly constant or decreasing evolution with redshift. Finally, we provide a fitting function for the galaxy-galaxy merger rate which is accurate over a wide range of stellar masses, progenitor mass ratios, and redshifts.