We present the results of a series of empirical computations regarding the role of major mergers in forming the stellar masses of modern galaxies based on measured galaxy merger and star formation histories from z 6 0.5 to 3. We reconstruct the merger history of normal field galaxies from z 6 3 to z 6 0 as a function of initial mass using published pair fractions and merger fractions from structural analyses. We calibrate the observed merger timescale and mass ratios for galaxy mergers using self-consistent N-body models of mergers with mass ratios from 1:1 to 1:5 at various orbital properties and viewing angles. We use these simulations to determine the timescales and mass ratios that produce structures that would be identified as major mergers. Based on these calculations, we argue that a typical massive galaxy at z 6 3 with M sub(*) > 10 super(10)M sub( )undergoes 4.4 super(+) sub(-) super(1) sub(0) super(.) sub(.) super(6) sub(9) major mergers at z > 1. We find that by z 6 1.5 the stellar mass of an average massive galaxy is relatively established, a scenario qualitatively favored in a -dominated universe. We argue that the final masses of these systems increase by as much as a factor of 100, allowing Lyman break galaxies, which tend to have low stellar masses, to become the most massive galaxies in today's universe with M > M*. Induced star formation, however, only accounts for 10%-30% of the stellar mass formed in these galaxies at z< 3. A comparison to semianalytic models of galaxy formation shows that cold dark matter (CDM) models consistently underpredict the merger fraction, and rate of merging, of massive galaxies at high redshift. This suggests that massive galaxy formation occurs through more merging than predicted in CDM models, rather than a rapid early collapse.