According to this link (now dead):
In the solar neighborhood, the stellar density is about one star per
cubic parsec (one parsec is 3.26 light-years). At the Galactic core,
around 100 parsecs from the Galactic center, the stellar density has
risen to 100 per cubic parsec, crowded together because of gravity.
So we'd see about 100 times as many stars as we see now, and the nearest star would most likely be less than 1 light-year away (compared to 4.3 light-years for Alpha Centauri).
This Wikipedia article says the stellar density near the Sun is only 0.14 stars per cubic parsec; it doesn't give a figure for the Galactic core. (If somebody has more information, please comment or edit.)
According to this Wikipedia article, the total integrated magnitude of the night sky as seen from Earth is -6.5. Making that 100 times as bright produces a total magnitude of -11.5 (5 magnitudes is a factor of 100 in brightness). The maximum brightness of the full Moon is -12.92.
So even with 100 times as many stars in the sky, the total brightness would be substantially less than that of a full moon.
(This assumes that the average brightness of the core stars is similar to the average brightness out here in the Galactic suburbs.)
The stars might be more dense close to the center, but I don't think you'd really want to be closer than 100 parsecs.