The R136 cluster in 30 Doradus is the prototype "super star cluster," and the only example sufficiently close that its massive star content can be studied directly. We have used the Hubble Space Telescope to obtain spectra of 65 of the bluest, most luminous stars in R136 and find that the majority of these stars are of type O3, the hottest, most luminous, and most massive stars known. The total number of O3 stars in this one cluster exceeds the total number known elsewhere in the Milky Way or Magellanic Clouds. The highest luminosity stars found are O3 If*, O4 If+, O3 If/WN6-A, and H-rich WN stars, with masses in excess of 120 , the highest masses for which appropriate evolutionary tracks are currently available. In accord with de Koter, Heap, & Hubeny, we conclude that these WN stars must be core H-burning stars whose spectra are WR-like because of high luminosity, and we find that their individual luminosities are a factor of 10 higher than is normal for WN stars of similar type but are like those found in the Galactic cluster NGC 3603, which they also resemble spectroscopically. Our spectroscopy does include stars as late as B0 V and samples most stars in the core of the R136 cluster with masses >50 M. The spectroscopy has been combined with HST photometry to study the star formation history and initial mass function of the R136 cluster. The young age (<1-2 Myr) for the highest mass stars, combined with what was previously known for the intermediate-mass populations, suggests that the lower mass stars began forming 4-5 Myr ago and continued forming until the high mass stars formed, consistent with the paradigm in which the formation of massive stars shuts down further star formation in the molecular cloud. Despite the unique preponderance of the highest mass and luminosity stars ever seen, the initial mass function (IMF) is found to be completely normal, with a slope Γ = -1.3 to -1.4. The number of high-mass stars is in good accord with that predicted by the IMF of the intermediate-mass stars, suggesting that a Salpeter-like IMF holds over the mass range 2.8-120 within the R136 cluster. The fact that the IMF slope in R136 is indistinguishable from those of Galactic and Magellanic Cloud OB associations suggests that star formation produces the same distribution of masses over a range of ~200 times in stellar density, from that of sparse OB associations to that typical of globular clusters. The large number of O3 stars in R136 is then simply a consequence of its youth (<1-2 Myr) and its richness, suggesting that the upper mass "cutoff" to the IMF seen in OB associations may simply be the result of their sparcity.