Why is Euler's Gamma function the "best" extension of the factorial function to the reals? There are lots (an infinitude) of smooth functions that coincide with $f(n)=n!$ on the integers. Is there a simple reason why Euler's Gamma function $\Gamma (z) = \int_0^\infty t^{z-1} e^{-t} dt$ is the "best"?  In particular, I'm looking for reasons that I can explain to first-year calculus students.
The Bohr–Mollerup theorem shows that the gamma function is the only function that satisfies the properties 


$f(1)=1$;
$f(x+1)=xf(x)$ for every $x\geq 0$;
$\log f$ is a convex function. 


The condition of log-convexity is particularly important when one wants to prove various inequalities for the gamma function. 



By the way, the gamma function is not the only meromorphic function satisfying
$$f(z+1)=z f(z),\qquad f(1)=1,$$
with no zeroes and no poles other than the points $z=-n$, $n=0,1,2\dots$. There is a whole family of such functions, which, in general, have the form
$$f(z)=\exp{(-g(z))}\frac{1}{z\prod\limits_{m=1}^{\infty} \left(1+\frac{z}{m}\right)e^{-z/m}},$$
where $g(z)$ is an entire function such that
$$g(z+1)-g(z)=\gamma+2k\pi i,\quad k\in\mathbb Z, $$
($\gamma$ is Euler's constant). The gamma function corresponds to the simplest choice
$g(z)=\gamma z$.

Edit: corrected index in the product.
For me another argument is the convincing one.     

Consider the log of the factorial resp the gamma-function; this is for the integer arguments a sum of logarithms of the integers. Now for the interpolation of sums to fractional indexes (which is required for the gamma to noninteger arguments) there exists the concept of "indefinite summation", and the operator for that indefinite summation can be expressed by a power series. we find, that the power series for the log of the Eulerian gamma-function matches exactly that of that operator for the indefinite summation of the sum of logarithms.
I've seen this argument elsewhere; I thought it has been here at mse before (by the used "anixx") but may be it is at MO; I'm not aware of a specific literature at the moment, but I've put that heuristic in a small amateurish article on my website; in the essence the representation of that indefinite summation is fairly elementary and should be existent in older mathematical articles. See "uncompleting the gamma" pg 13 if that seems interesting.     

Conclusion: the Eulerian gamma-function is "the correct one", because it is coherent with the indefinite summation-formula for the sums of consecutive logarithms.