Beethoven's Pathetique Sonata gives us a beautiful example of a perfectly used augmented sixth chord. Let us look at measures 41-51. The analysis looks something like this:
IV I6/4 V7 I V4/2 I6 IV6 Fr6 V V4/2 V6/5/IV IV V I
As we see in the 6th measure (m. 46) we are graced with the appearance of an Fr6. We are assured of this being a French sixth by the presence of an f, which shows its face in the right hand of the piano. I say that this is a perfect example of an Fr6 because of the absolutely regular resolution of the augmented sixth. The A natural, which is in the soprano, resolves up to a B while the A flat in the bass resolves down to a B flat.
There is another crazy chord in measure 49, which is definitely not an augmented 6th chord. I decided that it has to be a V6/5/IV resolving to an IV with the B natural in the top acting as a passing tone and the D flat being a chord tone.
A wonderful little ditty, the Pathetique.