Extreme microlensing events, defined as events with maximum magnification Amax >~ 200, are a potentially powerful probe of the mass spectrum and spatial distribution of objects along lines of sight toward the Galactic bulge. About 30 yr-1 such events are expected for main-sequence sources with I0 < 19. For many of these it is possible to measure both a ``proper motion'' and a ``parallax,'' that together would yield individual mass, distance, and transverse-speed determinations of the lensing object. The proper motion is determined from finite-source effects when the lens transits, or nearly transits, the source. The parallax is determined by observing the difference in the light curve as seen from two Earth observatories separated by about 1 Earth radius, R⊕. The size of the parallax effect is ~AmaxR⊕/r∼e, where r∼e is the projected Einstein radius and can be of order 1%. Detection of candidate events requires a pixel-lensing search of the entire bulge once per day, preferably by at least two observatories on different continents. Follow-up observation must be carried out using optical/infrared photometry, with short (e.g., 1 minute) exposures on small (>~1 m) telescopes. Extreme microlensing observations toward the Large Magellanic Cloud do not appear feasible at the present time.