Symmetry

Consider the two functions $x^{y}$ and $y^{x}$ with y as a parameter. The curves have either 1 nor 2 common points in dependence on y. For only 1 point the curves must touch i.e. the following equation system is valid:

(1) $x^{y} = y^{x}$

(2) $y\cdot x^{y-1} = y^{x}lny$

The solution of this is x=y=e (Euler's number)!

Euler's constant, $e$ , is the only number that fits the bill. So, the answer is $\boxed{2.718}$