<s>[INST] Find $f(f(f(f(f(f(\cdots f(x)))))))$ $2018$ times 
I was given a problem to calculate $f(f(f(\dots f(x))))$ $2018$ times if $f(x)=\frac{x}{3} + 3$ and $x=4$. 

Is it even possible to be done without a computer?
 [/INST] Let's try to spot a pattern.
$$f(x)=3+\frac x3=\frac13(3^2+x)$$ $$f(f(x))=f\left(\frac13(9+x)\right)=3+\frac{9+x}{9}=4+\frac x9=\frac1{3^2}(3^3+3^2+x)$$ $$f(f(f(x)))=f\left(\frac19(36+x)\right)=3+\frac{36+x}{27}=\frac{13}3+\frac x{27}=\frac1{3^3}(3^4+3^3+3^2+x)$$ 
So iterating $2018$ times, we have $$f(f(f(\cdots f(x))))=\frac1{3^{2018}}(3^{2019}+3^{2018}+...+3^2+x)$$ or $$3^1+3^0+3^{-1}+...+3^{-2016}+\frac x{3^{2018}}\approx 4.5+\frac{2018}{3^{2018}}$$ Now since $3^{2018}$ is much bigger than $2018$, that term could be negligible so we can conclude that $$\boxed{f(f(f(\cdots f(x))))=4.5}$$
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