Sukhoi SU-47 (1997, 2 produced): Russian experimental fighter aircraft featuring an unconventional forward swept wing/canard configuration. Forward swept wings provide greater maneuverability over conventional wing shapes. A similar American aircraft is the earlier Grumman X-29, which was also experimental.
Antonov An-72 (1977): Soviet transport aircraft. The unconventional placement of the engines benefits from the Coandă effect (which I won’t pretend to fully understand) – the exhaust air flowing over the top of the wings increases lift, resulting in less runway distance required to take off.
Aero Spacelines Super Guppy (1965, 5 produced under both variants): Cargo aircraft made from a modified C-97J, designed to carry large payloads, such as other planes (NASA T-38’s shown).
Blohm and Voss BV-141 (1938, 13-28 produced): German WWII era reconnaissance aircraft, most striking for its asymmetric shape. The position of the cockpit allows for greater downwards visibility, which would be beneficial for the aircraft’s role. The much more conventional approach would be simply to use two engines, such as on the much more widely produced Focke-Wulf Fw-189.
NASA AD-1 (1979, 1 produced): Experimental aircraft built by NASA to test the viability of an oblique wing. The wing would start off normally, but could rotate once in flight.. The oblique wing reduces drag in a similar manner as variable sweep wings (F-111 shown), but the oblique wing has the benefit of reduced complexity (since there is only one moving joint as opposed to two), as well as reducing the shift in the centre of gravity and lift of the aircraft as the wing moves compared to a more typical variable sweep wing.
North American F-82 Twin Mustang (1945, 272 produced): The American military piston powered fighter aircraft, designed as a long range escort fighter. The F-82 was designed to escort B-29 bombers on missions ranging over 3200km as part of the planned invasion of Japan. Japan surrendered before any F-82’s entered service, although F-82’s did see active service during the Korean war.
Beriev Be-200 (1998, 9 produced): Russian amphibious firefighting, search and rescue, and maritime patrol aircraft. Mounting the engines above the wings greatly reduces the amount of water spraying into the engines. You can see this approach used on many other amphibious aircraft such as the Consolidated PBY Catalina.
Scaled Composites Proteus (1998, 1 produced): A one-off high-altitude high-efficiency aircraft designed by the legendary Burt Rutan (who could easily fill up a post of his own). The aircraft has an all composite air frame, greatly reducing its weight. The Proteus has set several altitude records, topping out at over 63,000 feet (double the typical cruising altitude of commercial airliners) and can fly at that altitude for ~18 hours.
NASA M2-F1 (1963, 1 produced): The M2-F1 was one of NASA’s many experimental unpowered aircraft built in the 60’s and 70’s to test the concept of lifting body aircraft, which typically lack wings and generate most or all of their lift from the airframe itself. These tests were mostly geared spacecraft re-entry. Being unpowered gliders, these test vehicles had to be towed into the air, including by a Cadillac convertible on at least one occasion. The SNC Dream Chaser is a modern example of such a lifting body, which is currently in competition for NASA’s Commercial Resupply Services 2 contract.