In this paper, we study the properties of Diophantine exponents $w_n$ and $w_n^{∗}$ for Laurent series over a finite field. We prove that for an integer $n \geq 1$ and a rational number $w>2n-1$, there exist a strictly increasing sequence of positive integers $(k_j)_{j \geq 1}$ and a sequence of algebraic Laurent series $(\xi _j)_j$ such that $\deg \xi _j =p^{k_j}+1$ and $$ w_1(\xi _j)=w_1 ^{∗}(\xi _j)=\ldots =w_n(\xi _j)=w_n ^{∗}(\xi _j)=w $$ for any $j \geq 1$. For each $n \geq 2$, we give explicit examples of Laurent series $\xi $ for which $w_n(\xi )$ and $w_n^{∗}(\xi )$ are different.