  Extreme microlensing events, defined as events with maximum magnification
$A_\max\gsim 200$ are a potentially powerful probe of the mass spectrum and
spatial distribution of objects along lines of sight toward the Galactic bulge.
About 75 yr${}^{-1}$ such events are expected for main-sequence sources with
$I_0<19$. For many of these it is possible to measure both a ``proper motion''
and a ``parallax'' which 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, $R_\oplus$. The size of the parallax effect is $\sim A_\max
R_\oplus/\tilde r_e$ where $\tilde r_e$ is the projected Einstein radius. This
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
using optical/infrared photometry, with short (e.g.\ 1 minute) exposures on
small ($\gsim 1$ m) telescopes. Extreme microlensing observations toward the
Large Magellanic Cloud do not appear feasible at the present time.

Astrophysics