While there are compact cameras with fast lenses (like Lumix DMC-LX7, F1.4) they seem to be rather thick. The lumix is quite thick, at 46mm, largely due to the lens protrusion. Is it possible to make a thin compact camera with a fast lens? If not, what are the technical limitations? If it is possible, are any examples currently on the market?
As long as you understand that your size constraint is limiting how much quality you can have, the one you are looking for is the Canon Powershot S110. It measures 27mm at the thickest point and yet has both a slightly larger sensor than usual and much brighter F/2 aperture at wide-angle, only it goes down F/5.9 on the telephoto end. As a bonus, it is extremely well designed and very easy to use.
EDIT: The question is entirely different now. However, I believe the existence of the Canon S110 says it is possible. Now possible does not mean simple and I am sure that Canon has to overcome challenges to make it work.
Furthermore, products are rarely designed on one specification alone. Considering the maximum aperture is one criteria but it interacts with other. One could probably make an ultra-compact with an even brighter lens by removing the zoom or making the sensor smaller. Either case would cause issue with some users. What you really want is a large aperture and a large sensor and most people also want a decent zoom.
You can check it out by searching for a fast aperture among cameras. As you can see, only 4 models currently are ultra-compact. Two of those are the Canon S110 and S100 with slightly larger sensors than usual. The others have smaller sensors and show lower image quality.
Sensor size is especially significant for low-light shooting.
The Nokia 808 PureView is a cameraphone with a uniquely large sensor.
The sensor is 1/1.2" so it's close to that of the Nikon 1-series and the Sony RX100, yet the chassis is the size of a mobile phone. This sensor is far bigger than compact camera and mobile phone sensors usually are.
It is available new for about $350 if you look around.
In the last few days I heard that Nokia will be producing a Windows Phone 8 with a comparable sensor, though that could still be a year from release.
Search DPReview.com for more info.
I recently got a samsung ex2f which opens up to f1.4. I am quite happy with it. I dont have it with me right now, but specs says it measures 112.1 x 62.4 x 27.55mm (with the lens retracted), so it is within what you want.
Have to agree with the comments a bit subjective since new/better things are coming out but here's a camera that could match.
You didn't say anything about budget so.... Leica X2. 24mm lens f/2.8, ASPH sensor, good range of ISO 100-12500. It looks pretty small and handy to carry around.
Apparently a few people do think it's pretty good.
Also included with the purchase of X2, a copy of Adobe Lightroom 3/4 :D