Jan Philips writes:

Some of the early US manned orbital flights orbited only a little more
than 100 miles up. How low can you orbit? What is the relationship
between height and the maximum number of orbits? For instance, if you
go into a circular orbit at 100 miles, how many orbits can it stay in
orbit? Can you orbit at 95 miles? Etc.

Bear with me, I'll answer most of your questions.

First, there is no simple answer to how low can you orbit, although 100
miles (and note, every time I say miles I mean statute miles, which is
what you meant too. The only reason I feel I have to mention this is
that NASA, in its quest to use units no one in their right mind would
use, often expresses heights in nautical miles (AND statute miles AND
feet AND (but rarely) kilometers). When you are in an elliptical orbit,
force applied at the perigee (low point) tends to modify your apogee
(high point) and vice versa. The effect of this, for elliptical orbits,
is that they tend to become circular, as the larger friction force at
perigee lowers the apogee much quicker than the smaller friction force
applied at apogee (this is, of course, a simplification, as friction
operates all through the orbit, not just at apogee and perigee). Then,
once the orbit becomes circular, it tends to stay pretty circular,
although it is really spiralling in faster and faster. It turns out
that a 100 mile high circular orbit will last about a day, give or take
hours (this number varies because the density of the atmosphere varies,
mostly due to solar activity, and I might be off about a day, but it's
the right order of magnitude).

So, some examples. John Glenn's Friendship 7 entered a 99x165 mile
orbit, and he was told he was "go for 7 orbits". That doesn't mean that
he would have reentered after 7 orbits, just that the ground was certain
he wouldn't enter any earlier than that, so there was no need to worry
about completing his 3 orbit mission. Valery Bykovsky's Vostok 5
entered a 108x138 mile orbit, which was too low for its planned 8 day
mission, and by 5 days into the flight had decayed into an orbit below
100 miles circular and had to be brought down.

Some earlier US spy satellites would lower themselves from an orbit
about 400 miles circular to one of about 75x400 miles for an orbit or
two to get that much close to the target they were filming. They would
have to get out of that orbit quickly to avoid reentering and/or burning
up.

The shuttle's so-called direct insertion trajectory puts it into an
"orbit" with a perigee around 30 miles and an apogee above 200 miles.
This is not an orbit that will last even twice around the earth, which
is the reason for the OMS burn performed at first apogee to raise the
perigee safely out of the dense atmosphere. Even the reentry burn
doesn't really take a spacecraft out of orbit; it lowers the perigee so
low (around 20 - 30 miles) that the frictional losses quickly do the
rest and take the spacecraft out of orbit.