  Gravitational lensing can amplify the apparent brightness of distant sources.
Images that are highly magnified are often part of multiply-imaged systems, but
we consider the possibility of having large magnifications without additional
detectable images. In rare but non-negligible situations, lensing can produce a
singly highly magnified image; this phenomenon is mainly associated with
massive cluster-scale halos (>~1e13.5 Msun). Alternatively, lensing can produce
multiply-imaged systems in which the extra images are either unresolved or too
faint to be detectable. This phenomenon is dominated by galaxies and lower-mass
halos (<~1e12 Msun), and is very sensitive to the inner density profile of the
halos. Although we study the general problem, we customize our calculations to
four quasars at redshift z~6 in the Sloan Digital Sky Survey (SDSS), for which
Richards et al. (2004) have ruled out the presence of extra images down to an
image splitting of 0.3" and a flux ratio of f=0.01. We predict that 9-29% of
all z~6 quasars that are magnified by a factor of mu>10 would lack detectable
extra images, with 5-10% being true singly-imaged systems. The maximum of 29%
is reached only in the unlikely event that all low-mass (<~1e10 Msun) halos
have highly concentrated (isothermal) profiles. In more realistic models where
dwarf halos have flatter (NFW) inner profiles, the maximum probability is ~10%.
We conclude that the probability that all four SDSS quasars are magnified by a
factor of 10 is <~1e-4. The only escape from this conclusion is if there are
many (>10) multiply-imaged z~6 quasars in the SDSS database that have not yet
been identified, which seems unlikely. In other words, lensing cannot explain
the brightnesses of the z~6 quasars, and models that invoke lensing to avoid
having billion-Msun black holes in the young universe are not viable.
