What will limit the speed at which one can play the piano? I thought about this as a fun question.  Suppose we provide a piano player that is limited only by the typical relativistic rules (i.e., hands and fingers cannot move faster the speed of light), so choose your favorite speedy superhero or android.
In this scenario, what would limit the speed at which the player could play a typical (i.e., not electronic) piano?  Would it be:


*

*The strength of any of the following parts:


*

*piano keys?

*piano wires/strings?

*hammer arm?

*etc.


*The speed of the standing waves on the wires/strings in the piano?

*The delay between pressing a key down and the time when the hammer strikes the string?

 A: It will definitely be the inertia of piano action mechanism. The fact (or construction problem), that you can't repeat a tone fast enough is a real evergreen throughout the history of keyboard instrument development (read more here). Usually the trade-off for lower repetition rate is higher sensitivity of the key.
Sci-fi appendix: But right, let's suppose that we have a superhero or a very good player (i.e. real world superhero) with unlimited repetition rate (perhaps some electronic piano). Then the real problem in human percieving would be that really fast repeated tones can become a new tone with low frequency - we start to percieve fast pulses as a tone (not pure sine, of course). That is easy to sample e.g. in matlab. Try it! It's fun. For this phaenomena and more inspiration I would recommend you to listen the end of Nancarrow's Canon X
A: Let the hammer that strikes the string moves a distance $\Delta x$ above the string and then hit the string back due to gravity. Or take some other mechanism like a spring which puts the note back to the unpressed position after pressing that note. The time it takes to get a particular note back to unpressed position after it being pressed is the main factor here. Because for the artist to play something useful he'd need to repress a recently used note. 
The problem is not that we can't pluck the string or unpress a note with a speed faster than $c$. The problem is that we can't even play the piano faster than speed of sound in metal. The information of pressing the note reaches the string with the speed of sound in the materials involved. Let's take the distance between the note and the hammer $\Delta y$ then  if we try to press a note twice in a time $\Delta t$ then we have to make $ (\Delta x + \Delta y) v_{sound}< \Delta t$. Otherwise we'd just press the already pressed note. So if a tone involves pressing of a note twice in a time period then we can't make that time period smaller than $\Delta t$.
There might be further restrictions based upon how fast the string vibrates. A vibrating string is actually a standing wave. For a note to be heared the string must at least oscillate once. That is the wave must go from start to the end of wire at least twice. All this depends upon the speed of wave. This speed depends upon the length, weight, meterial and tension in the string. None of these factor involve $c$. Because the speed of wave is way too less than $c$. 
If your superhuman piano player also has a superpiano which is made up of super material which can transmit waves in strings with extraordinary speed then in principle the information of the note being pressed can't reach the string faster than $c$. Or if a guitar is being used then the speed of wave on string can't exceed $c$. If the length of the string of guitar is $L$ then you can't pluck a string twice in time $2L/c$
