Why are radians more natural than any other angle unit?

Consider the Taylor series for the trigonometric function. For instance sine $$ \sin \alpha = \alpha - \frac{\alpha^3}{3!} + \dots = \sum_{n=0}^\infty (-1)^{n}\frac{\alpha^{2n+1}}{(2n+1)!},$$ or cosine $$ \cos \alpha = 1 - \frac{\alpha^2}{2!} + \dots =\sum_{n=0}^\infty (-1)^n \frac{\alpha^{2n}}{(2n)!}.$$

If you were to choose some other unit for angle these very tidy series would pick up some additional factors in every term.

That kind of thing is "unnatural" to mathematicians.

Most importantly $$ e^{i x} = \cos x + i \sin x$$ only holds (in this form) in radians.

So now you might ask why $e$ is more natural than any other number ;-)

People call things "natural" when they simplify formulas.

Example, if there is a spinning wheel, the velocity $v$ of a point on the periphery is intuitively proportional to rotational speed $\omega$ and radius $r$. If the rotational speed is measured in radians per second, then the exact formula and the intuitive one are identical:

$$v = r \omega$$

rather than something ugly like $r\omega(\pi/180)$.