Triangular numbers can be discovered by taking any number $n$, and adding
$$\sum_{i=0}^n i = n + (n - 1) + (n - 2) ... 1 = \frac{n(n + 1)}{2}$$
These numbers can be generalized by putting any real argument in the expression above. Factorial is the function which is equivalent to multiplying
$$n(n - 1)(n - 2)...2*1$$
for whole numbers, and it can be generalized with the gamma function. Is there an a name for the exponent analog to the factorial function:
$$f(n) = n^{(n - 1)^{(n - 2)^{.^{.^{.^{1}}}}}}$$
and if so, is there a formula for a smooth curve that passes through the points for the natural arguments and generalizes it for the reals?