So, let's dig up and shed some light on those obscure machineries that were built but never raced. Or semi-raced in some cases.
1981 Williams FW07D
Here with Alan Jones behind the steering wheel testing at Donington Park.
The FW07D was an experimental six-wheeled test car (4 driven rear wheels, and 2 non-driven front wheels) that was tested by Alan Jones on one single occasion.
1982 Williams FW08B
The FW08B was a six-wheeled (4 driven wheels at the rear and 2 undriven wheels at the front) variant that originated from the FW07D (also six-wheeled). It never raced. Patrick Head specifically said that the reason it was banned was because "someone in a FOCA meeting said it would drive up costs and cause chaos during pitstops". The FIA promptly limited the number of wheels for all cars to four, of which only 2 may be driven.
1981 Lotus 88
The Lotus 88 was an innovative ground effect Formula One car designed by Colin Chapman, Peter Wright, Tony Rudd and Martin Ogilvie of Lotus in an effort to maximise the downforce produced by ground effects cars.The 88 used an ingenious system of having a twin chassis, one inside the other. The inner chassis would hold the cockpit and would be independently sprung from the outer one, which was designed to take the pressures of the ground effects. The outer chassis did not have discernible wings, and was in effect one huge ground effect system, beginning just behind the nose of the car and extending all the way inside the rear wheels, thereby producing massive amounts of downforce.
More on the car here
1995 Lola T95/30
The Lola T95/30 is a Formula One motor racing car which was tested by Allan McNish during 1994 and early 1995. The car, however, was not raced during any Formula One races. The car was designed as a prototype ready for Lola's arrival in Formula One. Lola, instead of racing the car in the actual season, chose instead to test the car while a search to find a big-name sponsor was carried out as money inside the project dried up.
More on the car here
1995 DAMS GD-01
Here pictured with Érik Comas behind the steering wheel (probably the first and last time he sat there) at the car's launch in 1995 at the Circuit de la Sarthe in France.
More on the car here
1996 Dome F105
The Dome F105 was an unraced Formula One car designed and built by the Japanese motorsport constructor, Dome.
The F105 was launched at the Spiral building in Tokyo on March 18, 1996.
More on the car here
1997 Lola T97/30
Powered by a Ford ECA Zetec-R 3.0 V8, this car holds the "record" of participating only in the 1997 qualifying session where it failed. Bad - because it had the aerodynamics of a brick. Later that year
More on the car here
1999 Honda RA099
Here pictured with Jos Verstappen behind the steering wheel, it's probably the most normal car in this list that was capable to race.
The project was looking promising, with the prototype machine setting competitive midfield times in tests alongside better funded and more established F1 teams. However, the project was aborted indefinitely after Postlethwaite's death from a heart attack at one of the Jerez tests, and the Dallara built tubs were no longer required.
More on the car here
2010 Toyota TF110
The Toyota TF110 was an un-raced, prototype Formula One car designed by Toyota Racing for the 2010 Formula One season. The car had been designed, and two chassis produced before Toyota officially decided to pull out of Formula One at the end of the 2009 Formula One season. One chassis was damaged by former team principal John Howett, while the other was used for a shakedown test. Various teams have attempted to purchase the chassis.
More on the car here
If you have more examples (and there are others), feel free to post them.
