Hash functions to the power of two and three notation
Yes, these notations always seem to mean that H^n(x) (or $H^n(x)$ in mathematical notation) means that $x$ is the first input, and that the next outputs are a hash of the previous output:

H^0(x) = x           // just defined as x below
H^1(x) = H(x)
H^2(x) = H(H^1(x)) = H(H(x))
H^3(x) = H(H^2(x)) = ... = H(H(H(x)))
H^4(x) = ............... = H(H(H(H(x))))
...


so another way to define it is $H^n(x)=H(H^{n-1}(x))$ where $H^0(x)=x$. I've used uppercase H as that seems to be more common.

The construct seems to comply with the exponentiation property within abstract algebra as only the less strong power associativity is required for the function composition of the (hash) functions (beware - the answer on the math site requires some mathematical background).



Usually this kind of construction is used for key stretching; a method of calculating a key from a password (or other low entropy secret) that includes a work factor. In other words, both the user and an attacker are required to perform a number of operations, which makes attacking the password harder.