I will make a guess about what the question intended to ask. If I am told that my interpretation is not the intended one, this answer will be deleted.
Let $f(x)$ be a function, and in general define $f^{(n)}(x)$ by $f^{(1)}(x)=f(x)$ and $f^{(n+1)}(x)= f(f^{(n)}(x)).$
Let $g(x,n)=f^{(n)}(x)$. I interpret the question as asking whether one ever is interested in $\lim_{n\to\infty}g(x,n).$
A quick answer is yes, often, this is a very important kind of question, with many applications. You will find a guide to a possible exploration in the following Wikipedia article. The iteration of functions, and the possible limiting behaviour, is a frequent theme in many branches of mathematics, both pure and applied.
Your specific question: Indeed, if we let $f(x)=\sin x$, and interpret "sequence of function compositions" as I did, the limit, as you conjectured, is $0$ for all $x$. But the situation with $f(x)=\cos x$ is different. You can explore this by putting your calculator into radian mode, starting at some number, and pushing the cos button repeatedly.