Is there official notation to represent "perform an operation n times"? I would like to know if you can represent the idea of say,

$$n^n$$

n amount of times without defining a function.  For example:

$$1$$
$$2^2$$
$$3^{3^3}$$
$$4^{4^{4^4}}$$
$$5^{5^{5^{5^5}}}$$

and so on, n amount of times.  Is there a way to represent this mathematically without defining a function?

EDIT: I understand that in this particular example the answer is tetration.  However, I am searching for more general notation that could also work with the following:

$$\ln(n)$$
$$n!$$
$${^n}n$$

all n amount of times.  Another example with factorial:

$$1!$$
$$(2!)!$$
$$((3!)!)!$$
$$(((4!)!)!)!$$
$$((((5!)!)!)!)!$$
It sounds like you are looking for the notation for an iterated function.  If $f$ is the function you want to iterate, you generally write
$$f^n$$
for the $n^\text{th}$ iterate of $f$.  (Notice that this is different from the notation for the $n^\text{th}$ derivative of $f$, which is given by $f^{(n)}$).  So, for example, you  could write
$$f(n) = n!$$
and then
$$f^1(n) = n!$$
$$f^2(n) = (n!)!$$
and so on.
You can use Knuth's up-arrow notation:

$a\uparrow n=a^n$

$a\uparrow \uparrow n=a^{a^{a^{\dots}}} $ n times

$a\uparrow \uparrow \uparrow n= a\uparrow \uparrow a\uparrow \uparrow a\uparrow \uparrow \dots$ n times

and in general

$a\uparrow ^b n=a\uparrow ^{b-1} a$ iterated n times