  Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the
rational function field $K:=\mathbb{F}_q(\theta)$. For a Drinfeld module $\phi$
defined over $K$, we study the transcendence of special values of the Goss
$L$-function attached to the abelian $t$-motive $M_{\phi}$ of $\phi$. Moreover,
when $\phi$ is a Drinfeld module of rank $r\geq 2$ defined over $K$ which has
everywhere good reduction, we prove that the value of the Goss $L$-function
attached to the $(r-1)$-st exterior power of $M_{\phi}$ at any positive integer
is transcendental over $K$.
