It's equivalent to show that there is no polynomial relationship f({2n choose n}, n!) = 0.  On the other hand, we know that {2n choose n} ~ 4^n/sqrt{n} asymptotically and n! grows much faster.  

Terence Tao once remarked that if a sufficiently simple duplication formula were known for the factorial then Wilson's theorem would give an efficient primality test.  (Edit:  see the other answer.  I may be misremembering the stronger remarks that Dick Lipton made.)