In this Letter, we report on the experimental generation of high energy (10 GeV), ultrashort (femtosecond-duration), ultrahigh current (∼0.1 MA), petawatt peak power electron beams in a particle accelerator. These extreme beams enable the exploration of a new frontier of high-intensity beam-light and beam-matter interactions broadly relevant across fields ranging from laboratory astrophysics to strong field quantum electrodynamics and ultrafast quantum chemistry. We demonstrate our ability to generate and control the properties of these electron beams by means of a laser-electron beam shaping technique. This experimental demonstration opens the door to on-the-fly customization of extreme beam current profiles for desired experiments and is poised to benefit a broad swath of cross-cutting applications of relativistic electron beams.