How many miles would a space shuttle have to go above earth to be completely unaffected by earth's gravity?

There is no possible way.  Every single object that has mass exerts gravity on every other object in the universe.  So the shuttle would always have gravity acting upon it, it just keeps lessening as it travels farther away, but it will never reach zero gravitational force.  This is also known as the law of universal gravitation.

Infinite.

Gravity affects everything no matter what the distance.

A space vehicle can orbit the planet at any altitude, even very low (as long as it doesn't hit a mountain). In orbit, a person would be in what is called "freefall" and would not feel the effects of gravity.
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Not possible.
Everything is always affected by the Earths gravity. There is  no such thing as zero gravity.
No matter how far you go from the Earth, There will always be a little gravity tugging on you.
Even if your 13.7 billion light years, There will still be some of Earth's gravity tugging on you, even if it's 0.0000000000000000001%

It's impossible to completely escape the Earth's gravity. Every object in the Universe affects every other object with its gravity. However, as you get farther away, the gravitational force will keep decreasing until it wouldn't really affect the shuttle anymore. It would still be there, but the force would be so tiny that it's like it's not even there anymore. Therefore, if the shuttle travels far enough away from Earth, it will eventually get to the point where Earth's gravity will be too weak to have an effect on the shuttle. In other words, it's possible to be "completely unaffected" by gravity when it gets too weak to influence an object, even though it's still there.

First off, in short, I'm thinking what you want to know about is escape velocity..which has little to do with altitude.

See...There is a way to travel along a path that isn't straight, but looks straight compared to, say, elliptical orbits. That is a hyperbolic escape trajectory, which depends on the mass of a planet to an extent, but also depends on the velocity of the space craft. A hyperbolic escape trajectory means that essentially, the space craft is on a path that will allow it, or has allowed it to escape the influence of the planet's gravity for all intensive purposes. The equation for the escape velocity is:

v >= sqrt(2GM/r), where r is the distance from the planet, G is the gravitational constant (which can be looked up), M is the mass of the planet.

For earth, a space craft has to be traveling at about 11 km/s to eventually escape, REGARDLESS of its altitude. If it continued at that speed, and the Earth was the ONLY body in the universe, eventually you would be able to just keep going forever. HOWEVER because there are other bodies in the universe, they exert influence on the space shuttle and so the shuttle's path will eventually be perturbed if it continues at this speed.

Now if that answer was completely irrelevant to you, I'm guessing what you mean is "what orbital altitude would the shuttle have to be at in order for gravity to be negligible...aka. At what orbital altitude can you consider the shuttle to be in a 'zero-gravity' environment?"

As the others have said, earth's gravitational influence extends to infinity. You might be wondering then about the earth's sphere of influence, which extends until the sun's influence takes over. In order for the ship to be affected negligibly by earth, the sun, or the solar system, you would have to travel deep into interstellar space, at least 2 light years away from the sun, between the sun and its closest neighboring star. That is the closest you could get to 0 g because it's the location where both gravitational fields of the sun and the star have minimal influence on the shuttle. If that's not good enough, the shuttle would have to venture into intergalactic space, somewhere between the milky way and its nearest galaxy neighbor. And even then, there would still be a bit of gravitational influence on the shuttle.
So in short, for all gravity to be negligible, at least for a short period of time, you'd have to travel pretty far.

However, on a simpler note, you can approximate the altitude at which Earth's gravitational acceleration in particular exerts a certain amount of influence. There is a simple equation you can use to estimate the gravitational acceleration at a particular altitude.

g = g0 (re / (re + h) )^2

re  = radius of earth (I typically use 6378 km)
h = orbital altitude
g0 = acceleration due to gravity at earth's surface (aka. 9.81 m/s^2)

The smaller g is, the less of an influence Earth's gravity will have on someone jumping out of the shuttle.
You could try solving the equation for h for some really small g...like 0.00000000001 percent of g0 and see what you get. Keep in mind though, that if the earth's sphere of influence still dominates at that altitude, if you jump off the shuttle, you will still float back into earth orbit.
For comparison, the international space station orbits between 280 and 460 km. Gravitational acceleration there is an average of 90% what it is on earth's surface. In contrast, g = 2.5 x 10^(-7) m/s^2 about halfway between the sun-centered orbits of earth and mars. That's really small.

Without any other objects in the universe, it would be about half a light year..?