Wonderful! Clearly, as you throw the pebble harder and harder, it goes higher and higher. If you throw it extremely hard, it will eventually escape the Earth's pull. That's what NASA does: launch objects with a lot of energy.
It turns out that for a spherical body (Earth is almost spherical), one parameter is enough to determine whether an object can escape the Earth's gravitational pull or not.
It is called the escape velocity. This is the minimum speed with which an object must be thrown vertically upwards for it to leave the Earth.