We're well familiar with swing wings, on small and large aircraft from both the east and west.

What other types of variable geometry have been tried?

Extending wings

The idea is to have a larger wing area at takeoff for more lift and a smaller wing area in flight for more efficiency.

The NIAI RK and followup RK-I used two tandem wings that served as rails for an extendable panel that could be rolled out between them.

The project failed because Stalin was so enthusiastic about it he had it use the most powerful engine available which was too unreliable. I can't find a reason why the concept was not tried again.

The German FS-29 glider of 1972 had a different arrangement. It had an outer wing that fit over an inner wing and could unsheathe like a sword. Only one was built.

Oblique Wing

A normal wing at takeoff then pivots during flight. High-risk idea to make transport aircraft more efficient in the transonic regime.

Lighter and simpler than a swing wing, with no change in the center of lift as the geometry is changed. Disadvantage is that flight characteristics become asymmetric left and right, plus problems with rigidity. See this question for more.

Variable Camber Wing

From the late 60s to the early 90s NASA tested experimental variants of their F-111s; at one period they were trying out a "Mission Adaptive Wing":

The second phase called transonic aircraft technology (TACT/F-111A)
added an highly efficient supercritical wing and later the third phase
applied advanced wing (Mission Adaptive Wing-MAW) flight control
technologies and was called Advanced Fighter Technology Integration
(AFTI/F-111A). source

The F-111 was already a swing-wing aircraft, but this modification was a supercricital mission-adaptive wing with smooth variable camber, similar to the aero-elastic wing already mentioned.

Flight research concept can be read here and results can be read here.

source In flight - compare with landing wing below.

source

In addition to types already mentioned by other answers:

Tiltwing

Was used to allow VTOL operations by tilting the entire wing, as can be seen on the Hiller X-18. The concept was never used outside testing as far as I can tell.

Aeroelastic Wing

Tested on the X-29 and later on the Boeing X-53, which was based on the F/A-18 Hornet. The idea here is that the wing can be twisted to control roll, giving better control while reducing load on the aircraft. Only used in testing so far.

Canard Rotor/Wing

The concept was that an aircraft could use a rotary wing similar to a helicopter for vertical take-off and landing; once up to speed it would stop the rotor and use it as a conventional wing. Was never tested in VTOL mode and the project was cancelled. See Boeing X-50 Dragonfly for more information.

Variable geometry wingtip

The XB-70 Valkyrie had hinged wingtips which could be angled downwards by up to 65 degrees to improve lift and stability in certain regimes.

Variable incidence, on the F-8 Crusader.

From the Wiki page:

The most innovative aspect of the design was the variable-incidence wing which pivoted by 7° out of the fuselage on takeoff and landing (not to be confused with variable-sweep wing). This allowed a greater angle of attack, increasing lift without compromising forward visibility.

Tiltrotor, on the V-22 Osprey

From the Wiki page:

A tiltrotor aircraft differs from a tiltwing in that only the rotor pivots rather than the entire wing. This method trades off efficiency in vertical flight for efficiency in STOL/STOVL operations.