Is there a mathematical term for raising a a number to the power of its own value.... eg. $5^5 , 1692^{1692}$.
Many Thanks
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$\begingroup$Adding a number to itself is a form of multiplication: $x + x = 2x$. Multiplying a number by itself is a form of exponentiation: $x\cdot x = x^2$. Raising a number to its own power is a form of tetration: $x^x = x\uparrow\uparrow2$.
$\endgroup$ $\begingroup$Both addition, multiplication, exponentiation and the operation you are describing are particular cases of the following recursive function $H: \mathbb{N}^3 \to \mathbb{N}$ (called "hyperoperation"):
$$H(n, a, b)= \begin{cases}b+1 & \quad n=0 \\ a & \quad n = 1 \\ 0 & \quad n = 2, b = 0 \\ 1 & \quad n \geq 3, b = 0 \\ H(n-1,a, H(n,a,b-1)) \end{cases}$$
It is not hard to see, that
$$H(1, a, b) = a + b$$
$$H(2, a, b) = ab$$
$$H(3, a, b) = a^b$$
and $H(4, a, b)$ is the exact binary operation on $a$ and $b$ of iterated exponentiation. That is the reason, why that operation is called "tetration" (from the Greek word "τέταρτος", which means "fourth").
So $a^a = H(4, a, 2)$ is the thing you were most likely looking for.
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