This paper was concerned with the Cauchy problem of the 3D magnetohydrodynamic (MHD) system. We first proved that this system was local well-posed with initial data in the Besov space ${\dot{B}}_{p,q}^s({\mathbb{R}}^3)$, in the critical Besov space ${\dot{B}}_{p,q}^{-1+\frac{3}{p}}({\mathbb{R}}^3)$, and in $L^p({\mathbb{R}}^3)$ with $p\in ]3,6[$, respectively. We also obtained a new growth rate estimates for the analyticity radius.