A small submarine, the bathyscape Trieste, made it to 10,916 meters (35,813 feet) below sea level in the deepest point in the ocean, the Challenger Deep in the Marianas Trench, a few hundred miles east of the Philippines. This part of the ocean is 11,034 m (36,200 ft) deep, so it seems that a submarine can make it as deep as it's theoretically possible to go. The water pressure at this depth is over 1000 atmospheres. Life does exist here, as well as a carpet of diatomaceous material that covers all the ocean floors of the world.
Trieste was manned by two people and funded by the United States Navy. The pressure sphere used was 2.16 m (6.5 ft) across, with steel walls 12.7 cm (5 inches) thick, able to withstand 1.25 metric tons per cm2 (110 MPa) of pressure. The pressure sphere of Trieste, which weighed 8 metric tons in water, was not neutrally-buoyant because the steel had to be so thick for a 2 m-sized sphere at that depth to withstand the pressure that it would have sunk like a rock on its own. Therefore Trieste's pressure sphere had to be attached to a series of gasoline floats, accompanied by iron pellets for weight.
Initially weighing slightly more than water, the craft descended 10.9 km (6.77 miles) below sea level. At the bottom, the pellets were ejected, and the buoyant gasoline floats carried Trieste back to the top. This feat has never been replicated by a manned craft, although several unmanned submersibles have since explored the Challenger Deep.
The deepest-diving large, military-style submarine was the Soviet submarine K-278 Komsomolets, with a hull made of titanium, making it very expensive, but able to withstand significantly deeper dives than the best submarines made of high-grade steel, like American nuclear submarines. The Komsomolets was a nuclear powered submarine specially designed to make trips as far down as 1300 meters (4265 feet) below sea level, definitely less than the Trieste, but very significant because the Komsomolets had to "defend" a much larger air bubble against the encroaching pressure of the surrounding ocean.
Compared to the best American nuclear submarines, of the Seawolf class, Komsomolets had about 78% better diving capabilities. Seawolf submarines have an estimated crush depth of about 2400 feet (730 m). The Seawolf submarines are constructed of a high grade steel called HY-100, capable of withstanding 100 atmospheres of pressure. As a rule of thumb, the pressure increases by one atmosphere for every 10 m of descent.