$\lim_{x\to 0} \frac{1-\cos (1-\cos (1-\cos x))}{x^{a}}$ is finite then the max value of $a$ is? I know the formula $\frac{1-\cos x}{x^{2}}=\dfrac{1}2$
But how do i use it here and tried l hospital and no use and doing l hospital twice will make it very lengthy how do i approach ?

You could try to approach this problem using Taylor series:
$$\text{cos}(x) = 1-\frac{x^2}{2}+O(x^4)$$
So 
$$\text{cos}(1-\text{cos}(x)) = \text{cos}\left(\frac{x^2}{2}+O(x^4)\right) = 1-\frac{\left(\frac{x^2}{2}+O(x^4)\right)^2}{2}+O(x^4)$$
$$ = 1-\frac{x^4}{8}+O(x^4)$$
And so on...
