On the one hand, there are at least $2^\mathfrak{c}$ sigma algebras on $\mathbb{R}$: one can take any subset $A$ of $\mathbb{R}$ and consider a sigma algebra $\{\emptyset, A, \bar A, \mathbb{R}\}$

On the other hand, number of sigma algebras can be bounded as $2^{2^\mathfrak{c}}$ since a sigma algebra is a family of subsets of $\mathbb{R}$.

Let's assume generalized continuum hypothesis to simplify. In this case there are only two possible answers. Unfortunately I can't either find $2^{2^\mathfrak{c}}$ different sigma algebras, or prove their number is exactly $2^\mathfrak{c}$.