Of course, traditional piano literature does not extend the notation beyond the usual 88 keys.
The fascination I have for extra keys involves playing Busoni's or Liszt's transcriptions of J.S. Bach's organ recordings on the piano.
In addition, Busoni's transcription of JSB's Chaconne has, on the last page a series of descending lowest piano octaves ---- D, C, Bb, A, ... and proceeds to fall short on the final G, because one does not exist, save for Bosendorfers and Stuart & Sons instruments. When I perform the Bach-Busoni Chaconne, I (being the piano tuner) DE-Tune the lowest B natural down to low G!!! In that way, I can create a mini-Bosendorfer sound to the ending of the piece! Besides, B natural is not used in this piece (D minor wherein the B's are flatted.)
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The other reason I am so fascinated with extended keyboards ... involves the sonority of the piano. When the sustain pedal is depressed, all dampers are raised as expected. The raised-damper condition allows ALL strings whose natural resonant modes coincide with the regular notes being played ... to vibrate in sympathetic motion with the original notes.
The above paragraph addresses WHY electronic pianos sound so weak when playing with the sustain pedal depressed -- they only "sustain" the notes being depressed at the time, whereas a good piano's strings create a literal symphony of sound.
Here is a simple experiment for you to try on your own piano. It consists of two variations of the same sympathetic vibration theme.
Slowly depress C3, such that only its damper is released but the string is not vibrating. Whilst holding down C3, then hit the lowest C with a sharp attack, and listen to what happens. You should be able to hear the C3 pitch ring out, even though you did not hit it with the hammer.
Next try the same thing with G4, except strike the LOWEST C -- this time you will hear the G ring out, despite "only" having hit the lowest C note.
As a variant to the above, slowly depress the lowest C key, allowing its damper to release, and the string does not vibrate. Next hit a C3 or a G4 or any number of other tones, either singly or together. You will hear those very same tones in the "non-struck" low C. In reality, the low C string ALREADY CONTAINED those extra harmonics, but most people are not able to recognise, unless revealed by these party tricks!
Going one step further, it is this application of sympathetic vibration that adds to the tonal pallete of a gorgeous piano.