  OGLE-2004-BLG-343 was a microlensing event with peak magnification
A_{max}=3000+/-1100, by far the highest-magnification event ever analyzed and
hence potentially extremely sensitive to planets orbiting the lens star. Due to
human error, intensive monitoring did not begin until 43 minutes after peak, at
which point the magnification had fallen to A~1200, still by far the highest
ever observed. As the light curve does not show significant deviations due to a
planet, we place upper limits on the presence of such planets by extending the
method of Yoo et al. (2004b), which combines light-curve analysis with priors
from a Galactic model of the source and lens populations, to take account of
finite-source effects. This is the first event so analyzed for which
finite-source effects are important, and hence we develop two new techniques
for evaluating these effects. Somewhat surprisingly, we find that
OGLE-2004-BLG-343 is no more sensitive to planets than two previously analyzed
events with A_{max}~100, despite the fact that it was observed at ~12 times
higher magnification. However, we show that had the event been observed over
its peak, it would have been sensitive to almost all Neptune-mass planets over
a factor of 5 of projected separation and even would have had some sensitivity
to Earth-mass planets. This shows that some microlensing events being detected
in current experiments are sensitive to very low-mass planets. We also give
suggestions on how extremely high-magnification events can be more promptly
monitored in the future.
