Is there any use for $\sin(\sin x)$? The convention that $\sin^2 x = (\sin x)^2$, while in general $f^2(x) = f(f(x))$, is often called illogical, but it does not lead to conflicts because nobody uses $\sin(\sin x)$.
But is this really true? Or is there a real-world application in which $\sin(\sin x)$ occurs? Or maybe something a bit more general, like $\sin(C \sin x)$ for some constant $C \neq 0$?
 A: Just another cw, in way of an illustration: level plot for modulus and argument of $\sin(\sin(z))$ in the complex plane

(Thanks to Michael E2 from Mathematica.SE for helping out with the plot)
A: Valerii Salov, in Notation for Iteration of Functions, Iteral, has written a 23 page (!) paper on how to denote the iteration of a function, specifically for the issue raised by the OP in Inevitable Dottie Number. Iterals of cosine and sine.
The notation proposed by Salov uses the first Cyrillic letter И of the Russian word Итерация for iteration, as a variation on the product symbol $\prod$, so that И$_v^n f$ is the n-fold iteration of the function $f(x)$ for initial value $x=v$.
Some examples of the use of this compact notation:

This can be typeset in MathJax as a LARGE version of Unicode character 1048:
$$\large{
{\LARGE\unicode{1048}}_0^n(x+1)=n,\
{\LARGE\unicode{1048}}_{x=1}^n ax=a^n,\
{\LARGE\unicode{1048}}_{1}^\infty \frac{1}{x+1}=\lim_{n\to\infty}
{\LARGE\unicode{1048}}_{1}^n \frac{1}{x+1}=\frac{2}{1+\sqrt{5}}
}$$
A: The intensity of light diffracted at a slit as a function of the angle actually involves a term $\sin\left(\frac{\alpha\beta}{2}\sin(\theta)\right)$, see
https://en.wikipedia.org/wiki/Fraunhofer_diffraction
(I'm no physicist at all, but this has been stuck in my head since high school just because it is such an unusual term to encounter naturally)
A: Since $\sinh(x) = i\sin(i x)$ is the odd part of the exponential function, we can interpret it (for example within the framework of combinatorial species) as the (exponential) generating function for sets of odd size.
Thus, $\sinh(\sinh(x)) = -i\sin(\sin(ix))$ is the (exponential) generating function for set partitions with an odd number of parts, each of which has odd cardinality.
We can slightly refine this by interpreting $\sin(\omega\sin(x))$ as the generating function of a weighted species, giving each set partition the weight $(-1)^{n-1} w^b$, where $b$ is the number of blocks and $n$ is the size of the ground set.
Similarly, $\cosh(\sinh(x))$ is the generating function for set partitions with an even number of parts, all of which are of odd cardinality, and $\cosh(\cosh(x)-1)$ is the generating function for set partitions with an even number of (nonempty) parts, all of which are of even cardinality. Note however, that the coefficients of $\cos(\cos(x)-1)$ and $\cosh(\cosh(x)-1)$ are very different, whereas the coefficients of $\sin(\sin(x))$ and $\sinh(\sinh(x))$ only differ in sign.
As an aside, $\sin(\sin(\cdot))$ satisfies a nice differential equation:
$$
(f^2-1)^2 (f''' + f') - 3 (f^2-1) f f' f'' + (2f^2+1) f'^3 = 0
$$
while $\sinh(\sinh(\cdot))$ satisfies:
$$
(f^2+1)^2 (f''' - f') - 3 (f^2+1) f f' f'' + (2f^2-1) f'^3 = 0
$$
A: People in complex dynamics consider the behavior of all sorts of functions under iteration. For example, here is the Julia set of $\sin(z)$.

In that context, it makes perfect sense to talk about $\sin(\sin(\cdots \sin(z)))$.
A: A frequency-modulated (FM) signal (like those used in FM radio), with a sinusoidal input, can be represented as
$$x_\mathrm{c}(t) = A\cos(\omega_\mathrm{c}t+\beta\sin \omega_\mathrm{m}t), \tag{1}$$
where $\omega_\mathrm{c}=2\pi f_\mathrm{c}$ is the carrier angular frequency (e.g. $f_\mathrm{c}$ ranges from 87.5 MHz to 108 MHz for the FM radio), $\omega_\mathrm{m}$ is the modulating angular frequency (e.g. that of an audio tone) and $\beta$ is the modulation index (a normalized measure of the intensity of the modulating signal).
If you decompose (1) with the usual trigonometric identity for the sum of angles you get the term $\sin(\beta\sin \omega_\mathrm{m}t)$. This decomposition can be expanded in a Fourier series with coefficients generated by Bessel functions, thus obtaining the spectrum associated to the FM signal.
A: I think Carlo Beenakker's comment deserves to be upgraded to an answer:
Bessel functions have integral representations involving such iterates of trigonometric functions. For example,
$$
\int_0^1\sin(x\sin(\pi t))dt=\mathbf{H}_0(x)
$$
is the zeroth Struve function, one in the crowded family of various Bessel and Bessel-like functions.
A: The characteristic function for the Poisson distribution is given by
$$
e^{\lambda\left(e^{i\theta}-1\right)}=e^{\lambda\left(\cos\theta-1\right)}\left(\cos(\lambda\sin\theta)+i\sin(\lambda\sin\theta)\right)
$$
