The function $$f(x)=\cos(\sin(x))-\sin(\cos(x))$$ has a unique fix-point.
The solution of $\ f(x)=x\ $ is $$0.15328786038074973385826057\cdots$$
Is this number rational, algebraic irrational or transcendental ?
We cannot use the fact that $\cos(x)$ and $\sin(x)$ are transcendental for algebraic $x\ne 0$ because we have nested trigonometric functions and moreover a difference of such functions.
The continued fraction and the algdep-calculations with PARI/GP seem to indicate that the number is transcendental, but of course this is not a proof.