Question: If i have $f:\mathbb{R}\to \mathbb{R}$ and i know $f\circ f$ is differentiable then does that mean that $f\circ f\circ f$ is differentiable ?

I can't tell if i need to counter this or prove.
And if i have to counter then what kind of counter?

Answer: 
No : take $f$ to be an involution (i.e. $f\circ f(x)=x$) that is not differentiable. An example of such a function is

$$
f(x)=\begin{cases}
1 &\text{if $x=-1$} \\
-1 &\text{if $x=1$} \\
x &\text{otherwise} \\
\end{cases}.
$$

Notice how this function is not even continuous.
No. Take for example the function: 
$${f(x) = 
\begin{cases}
\frac{1}{x} &\text{if } x \neq 0 \\
0 &\text{if } x = 0
\end{cases}}$$

The function $f \circ f$ is continuous and differentiable , but $f \circ f\circ f$ isn't.