Is the notation "16va" or "16vb" ever used to mean to transpose what is written up or down two octaves? I cannot think of how else you would represent the highest and lowest notes on a piano without using excessive ledger lines, but I have never seen it written that way. Can you think of any pieces that use those parts of the piano's range? Do you know how that sheet music is written down?
First off 2 octaves above or below is a 15th because an octave is 7 letter named notes above a unison (P1) so to get the first octave you have 1 + 7 = 8. 7 more notes above that is the next octave so 8 + 7 = 15. However it is very rare because as you inferred, it is rare that a pianist will play that high up and also it is much easier to understand just the 8va or 8vb than 15ma or 15mb. Also from a practical standpoint the grand staff can comfortably fit 5 octaves from C2 to C6 which is equivalent to 60 keys on the piano. If you introduce notation for an octave above or octave below, you now cover 7 octaves from C1 to C7 which is equivalent to 84 keys. Since the normal max for a piano is 88 keys, you can see that having 8va or 8vb gives you access to pretty much every note within two ledger lines of the grand staff.
8va is the abbreviation for the Italian "all'ottava", or "at the octave (up)" while 8vb is the abbreviation for "all'ottava bassa" or "at the octave below".
The notation does exist, and you may see it when you have a part that repeats itself multiple times, each time an octave higher. I personally have never seen any song with that notion, but it does exist.
The double-octave transposition notation is indeed used in many kinds of classical and contemporary music. (The piano is only as big as it is to begin with because composers kept pushing the boundaries.)
Your idea is very close, but incorrect. The notation is actually 15ma or 15mb, because the number in question refers to the size of the interval.
| 1 2 3 4 5 6 7 | 8 9 10 11 12 13 14 | 15 | C D E F G A B | C D E F G A B | C
In doing further research about the terminology, I came across this post on the ABRSM forums:
This actually clears up some confusion I've had for a few years after seeing an 8va used on a bass clef to mean "down an octave". It seems in actuality the basso addition is relatively recent in the grand scheme of things, and so our current usage of these terms will not necessarily be consistent with older printings of music.