Let * be the binary operation on N defined by
       a * b = HCF of a and b.
Does there exist identity for this binary operation one N?

Let e be the identity element. Then,

$$\begin{gathered} a * e=a=e * a,\forall a\in N \\ \text{HCF }\left(a, e\right)=a=\text{HCF }\left(e, a\right),\forall a\in N \\ \Rightarrow \text{HCF }\left(a, e\right)=a,\forall a\in N \end{gathered}$$

We cannot find e that satisfies this condition.
So, the identity element with respect to * does not exist in N.