Answer:
Half.
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But the amount of it that you're ever able to see depends on where you live ...
specifically, your latitude, i.e. how far you are from the equator.
If you're at the north or south pole, then you see half of the entire sky whenever
it's dark enough. The center of it is directly over your head. Everything rotates
around that center, never getting any higher or lower off the horizon, and you
never see any of the other half. (By the way, since you're at the pole, you get
all of your dark time all at once. The sun is up for 6 months and down for 6
months, and it's probably dark enough to see stars for only maybe 4 or 5
months out of those six ... out of which 2 or 3 are too cold to go outside, or
the sky is overcast, or full of blowing snow. Astronomers do not pine for an
observatory at the north or south poles.) )
If you're on the equator, then you also see half of the entire sky whenever it's
dark enough. The center of it is on your northern (and southern) horizon.
Everything rotates around that center like a barrel, rising in the east, setting
in the west 12 hours later, and over the course of few months, you can see
everything in every part of the sky, at some time during the night.
If you're in between the equator and a pole, then you can see half of the entire
sky whenever it's dark enough. The center of it is somewhere on a slant between
your horizon and the point over your head. The angle between the horizon and
that center-point is the same angle as your latitude, and the whole sky turns
around that center.
Since the center-point is on a slant . . .
-- some things in the sky that are close enough to it never get low enough to set;
-- some things in the sky that are so far from it in the other direction never rise,
and you can never see them from where you are;
-- everything else in your sky, between these two circular areas, rises and sets.
The whole action shifts about 4 minutes earlier every night, so the parts that are
only up during the day right now will eventually be up during the night if you wait
a few weeks or months.