The bottomless pit is a concept sometimes found in video games, cartoons, fairy tales, and mythology. The basic idea is that of a hole in the ground that is so deep, it literally does not have a bottom to it. In cartoons, this is often demonstrated when a character drops a rock into the pit. The usual cartoon whistle of a falling object is heard, but instead of an eventual impact the whistle simply fades out. Optionally, the character may gulp nervously.
In video games, the bottomless pit is usually an automatic death for the player, regardless of whether or not he could normally sustain multiple wounds before dying. This effect can usually be duplicated with a pit of water, lava, or spikes just as well, although in rare cases some games differentiate between these by giving the player ways to survive some perils but not others (for example, learning to swim or being able to fly out of the empty pit). They are often found underneath "jumping puzzles", a series of stationary, moving, or vanishing platforms that must be crossed with a series of well-timed jumps to reach some goal. Players generally do not like jumping puzzles.
In cartoons, fairy tales, and mythology, a bottomless pit is sometimes a peril the hero must brave and occasionally the method by which he defeats his foe. Other times a bottomless pit is not possible because digging down far enough would reveal Hades, Hell, or some other type of Underworld. This underworld may, however, itself contain a bottomless pit as one of its sources of eternal punishment. In cartoons specifically, a portable hole often becomes a bottomless pit when placed on the ground.
A tunnel all the way through the Earth
Of course the literal idea of a bottomless pit on our spherical Earth is rather silly. Dig down far enough and you're either going to have to stop eventually, or you'll dig right through to the other side of the planet (somehow digging through all the magma and liquid nickel and iron in the process). Keep in mind that the deepest hole ever dug is less than halfway through the thin layer we call the Earth's crust (see footnote).
A hole dug all the way through the Earth would be a sort of bottomless pit in that, when viewed from either direction, there would be no literal bottom to it. However it fails on two points.
Point one is that an object dropped into a true bottomless pit would fall forever, while an object falling into such a hole in the Earth would eventually (after swinging back and forth like a pendulum for a very, very long time) come to rest at the center of the Earth, where the gravity from all sides cancels out to zero.
Point two is that a true bottomless pit should never be able to be filled, and this type of bottomless pit has a depth to it equal to the diameter of the Earth (about 12,700 kilometers).
A notable feature of a hole through the Earth is that due to the rotation of the Earth, the hole would have to be curved rather than straight down if an object is to fall freely through without impacting the sides. This is because an object on the surface has a tangent momentum to the East due to the rotation of the Earth. (A hole through the poles being an exception, but keep in mind the North pole is covered in water with a layer of floating ice, not dry land).
A black hole
A black hole seems to qualify as a bottomless pit, but only on a technicality. A black hole can never be "filled up" since all matter added to it just makes it "deeper". The black hole itself, however, is not infinitely large when viewed from the outside.
Should a true bottomless pit exist, the best use for it would be waste disposal. It would be a great place to dump toxic waste, high level radioactive material, and the occasional politician. Just remember that anything tossed in would be irretrievable, so don't dump anything you might want back later, say for example recyclables.
In a metaphorical sense, a bottomless pit is used to describe a situation or a problem that could indefinitely swallow up money, time, or resources without ever showing substantial improvement. When facing a political, military, financial, or business bottomless pit, it is usually better to cut your losses and quit than continue trying to solve the apparently hopeless situation. A classic example of a bottomless pit would be the 1986 Tom Hanks / Shelley Long movie The Money Pit, in which every attempt to fix one problem on the house only served to reveal two or more underlying problems behind it.
The world's deepest hole, by the way, is currently the Russian Kola Peninsula borehole at 12 km (40,000 feet). This is only about 38% of the way through the Earth's 32 km (20 mile) thick crust, and it was dug between 1970 and 1994.