Is there a formula to find the factorial of a number?

The factorial of a natural number (integers > 0) is the product of all natural numbers less than or equal to that number. It is denoted with as N!.
5! = 5*4*3*2*1 = 120

I am trying to figure out if there a known closed-form formula for the factorial operation. The formula must have a fixed number of operations, no matter how large the number is. Operations can include addition, multiplication, exponentiation, and any inverses for those (subtraction, division, roots, and logarithms). If you must, you can include things such as 3^X or even hyperoperations.

Can such a formula exist? If it can exist, does anyone know what it is?

I was inspired to ask this question by the fact that a similar-looking function 1+2+3+4+5=15 has a known formula, (N^2+N)/2.