We report on a systematic study of the most massive stars, in which we analyzed the spectra of four very luminous stars in the Large Magellanic Cloud. The stars lie in the 30 Doradus complex, three of which are located in the core of the compact cluster, R136a (R136a1, R136a3, and R136a5), and the fourth (Melnick 42), located about 8" north of R136a. Low-resolution spectra (<200 km s-1) of these four stars were obtained with the GHRS and FOS spectrographs on the Hubble Space Telescope. The GHRS spectra cover the spectral range from 1200 to 1750 A, and the FOS spectra from 3200 to 6700 A. We derived the fundamental parameters of these stars by fitting the observations by model spectra calculated with the "ISA-WIND" code of de Koter et al. We find that all four stars are very hot (~45 kK), luminous, and rich in hydrogen. Their positions on the HR-diagram imply that they are stars with masses in the range 60--90 M⊙ that are 2 million years old at most, and hence, they are O-type main-sequence stars still in the core H-burning phase of evolution. Nevertheless, the spectra of two of the stars (R136a1, R136a3) mimic those of Wolf-Rayet stars in showing very strong He II emission lines. According to our calculations, this emission is a natural consequence of a very high mass-loss rate. We conjecture that the most massive stars in R136a---those with initial masses of ~100 M⊙ or more---are born as WR-like stars and that the high mass loss may perhaps be connected to the actual stellar formation process. Because the observed mass-loss rates are up to 3 times higher than assumed by evolutionary models, the main-sequence and post--main-sequence tracks of these stars will be qualitatively different from current models. The mass-loss rate is 3.5--8 times that predicted by the analytical solutions for radiation-driven winds of Kudritzki et al. (1989). However, using sophisticated Monte Carlo calculations of radiative driving in unified model atmospheres, we show that---while we cannot say for sure what initiates the wind---radiation pressure is probably sufficient to accelerate the wind to its observed terminal velocity, if one accounts for the effects of multiple photon scattering in the dense winds of the investigated stars.