The F-Duct, and also the double diffuser in 2009.

Ross Brawn even warned the other teams late in 2008 that the 2009 rules were open to interpretation, and then wiped the floor with everyone in the first half of the season. By then everyone had a DD of course.

EDIT: Also the hot-blown diffuser, where the engine management programme pushed fuel through the engine off-throttle with retarded ignition, which didn't put power through the wheels but due to specially shaped exhausts pushed more gas through the diffuser to get more downforce in cornering.

The F-duct was a clever work around of the old F1 rule of no moveable aerodynamic devices (which dates from the 70's) and was motivated, intially by the practice of mounting the rear wing directly to unsprung parts of the suspension. A more applicable example of the rule was the late 70s brabham fan car which used a large fan at the rear (ostensibly for engine cooling purposes) to suck out air from the underside of the car to add downforce. Think of the hyperloop design, in reverse.

By making the only physical moving part in the f-duct design the driver (who obviously has to be allowed to move), the devices cleverly adhered to the letter of the law. The driver would close a hole in the cockpit which allowed some air to pass or not to a fluidic switch (think of a transistor), which would influence the flow of a larger stream of air to the rear wing).

One issue with banning the engine mapping that blew the diffuser off throttle was that this was already being done by many engine manufacturers for engine cooling purposes, and thus could not be easily regulated. Even their year end attempt at a fix by specifying the location, pointing direction, etc of the exhaust outlets didn't work (see coanda effect / downwash exhausts of 2012). They have finally addressed the solution in 2014 regulations by specifying that the exhaust outlet has to be behind all bodywork.

It's also worth pointing out that exhaust blown diffusers are more than a decade old, but in the past suffered from the problem of changing the balance of the car drastically mid throttle (when the driver gets on the throttle) due to 1) less advanced engine mapping), and 2) directly feeding into the diffuser venturis. In fact, for a long time, the standard solution (periscope exhausts pioneered by ferrari), sought to remove as much influence of exhaust air on the aerodynamics as possible to make the car's balance unaffected by engine exhaust volume, so given that environment, no one was worried about anyone using the exhaust for aerodynamic benefit. The modern (Red bull design) blown diffusers only channel a relatively small portion of exhaust air directly into the diffuser channel, and instead use a large portion of exhaust to blow over the top of the diffuser itself (enhancing the diffuser effect), and on the side of the diffuser (to provide a air skirt that prevents squirt air from the rear tires bleeding into the diffuser from the side which decreases efficiency), particularly on cars like the Red bull which run a high level of rake (which enhances aerodynamic downforce separately by placing the front wing closer to the ground but makes the diffuser more vulnerable to tire squirt and other effects).

I'll add that the no moveable aerodynamic devices ruling has been applied very widely, and in sometimes surprising ways. The prime example is the ban on Renault's tuned mass damper from the 2005 time frame. It was a mass damper inside the nosecone, which meant that the moving masses were never exposed to the air outside the car. It was ruled that by influencing the movement of the front wing (maintaining it at a more optimal height), it was therefore a moveable aerodynamic device. The sad part of this story is that the regulators didn't actually understand what benefit the device was supposed to provide, which was to allow for a more compliant suspension (a non aerodynamic advantage), and in fact Mclaren had a suspension element, the interter, which did effectively the same thing, but was not banned.

I know it'll sound like I'm just coming at the same thing from the other side, but the purpose of the mass damper was not to allow for more compliant suspension, but to stabilize the aero platform.

And the way the regs were written at the time, "any specific part ofthe car influencing its aerodynamic performance must be rigidly secured to the entirely sprung part of the car (rigidly secured means not having any degree of freedom)," and "must remain immobile in relation to the sprung part of the car".

The regulators fully understood its benefit–as did McLaren, Ferrari, Toro Rosso, Honda, and Midland, who were all testing their own versions of that system–but it's hard to see how a moving weight strapped to the sprung part of the car in order to improve the aero platform is anything but illegal.

That was precisely the misunderstanding. The purpose of the device was to improve mechanical grip by allowing the tires to have a more uniform pressure with the ground, removing some of the influence of impulses applied to the tire/suspension and the attendant lateral oscillations that result.

With such a rigid interpretation of the regulations, one could say the anti roll bar, spring, or damper would also be in violation of the regulations, as they have knock on effects concerning the aerodynamics.

It improved both mechanical and aero grip at the same time, but the primary purpose, as per Toyota's senior chassis engineer, was to be able to stabilize the aero platform, specifically around pitch. Which makes sense: same reason active suspension came into being, even at the expensive of grip. Aero gains are more productive, on balance, than mechanical grip gains, at least in Formula 1.

It was moveable ballast, plain and simple. In its early guise, was totally unconnected from the suspension. So that's a pretty easy distinction to make vs. springs/dampers/anti-roll bars.