how far away can something be from the earth and still be in orbit? Just as the title asks,
How far away can, say, a satellite be and still be in "orbit" ?
How about for a given velocity?
Fun Facts
200 miles (320 km) up is about the minimum to avoid atmospheric interference. The Hubble space telescope orbits at an altitude of 380 miles (600 km) or so.
potentially helpful numbers
mass of Earth = 5.97219 × 1024 kilograms
mass of the Moon = 7.34767309 × 1022 kilograms
distance (earth, moon) = 238,900 miles (384,400 km)
 A: In principle, arbitrarily far. The velocity decreases with the distance because the attractive gravitational acceleration has to match the centrifugal one,
$$ \frac{GM_\text{Earth}}{R^2} = \frac{v^2}{R},\quad v = \sqrt{\frac{GM}{R}} $$
for a circular orbit. So the velocity decreases with the distance as a power law but it never gets zero.
In practice, it becomes meaningless to talk about the "Earth's orbit" if the distance $R$ is comparable to the distance to other important celestial bodies such as the Sun because the gravity of those other bodies will act differently on the Earth and the "distant satellite" which will disturb the simple elliptic/circular orbits.
For example, if the distance $R$ becomes really comparable to 1 AU, the distance between the Earth and the Sun, the Earth's satellite will become mostly the Sun's satellite, of course. There are also special points, the Lagrange points, in which the Earth's gravity and the gravity from another body conspire so that the relative position of the Earth, the other celestial body, and the probe affected by their fields remains constant in time.
A: In the restricted three body problem, where you consider two objects orbiting each other, such as the sun and earth, and the motion of a third object that does not affect the movement of the first two, but is affected by their gravity, you can sort of figure out how far/fast from one object you have to be to not be orbiting it anymore.

The picture above is taken from Shane D. Ross' Ph.D. thesis. Depending on the total energy of the third mass, it will never be able to go into the shaded areas. So if you are orbiting Earth, which would be $m_2$ in the Sun-Earth example, or $m_1$ in an Earth-Moon one, there is a minimum energy at which can break out of the first and start orbiting the other body. The transition point is the Lagrangian point $L_1$. At a higher energy, it is possible to break away to infinity from both objects, the transition point corresponding to the Lagrangian point $L_2$.
So depending on a more precise definition of you question, a possible answer is that a satellite beyond the Sun-Earth L1 point is more orbiting the Sun than the Earth. The Sun-Earth L1 point is, according to this, about 1% of the way to the Sun. So that's about 1,500,000 Km. You could of course calculate the corresponding $E_1$ enery and translate that to kinetic energy and velocity.
