We study the boundary behaviour of a meromorphic transcendental map $f :
C \to \widehat{C}$ on an invariant simply connected Fatou component $U$. To
this aim, we develop the theory of accesses to boundary points of $U$ and
their relation to the dynamics of $f$. In particular, we establish a
correspondence between invariant accesses from $U$ to infinity, weakly
repelling points of $f$ and boundary fixed points of the associated inner
function on the unit disc. We apply our results to describe the accesses
to infinity from invariant Fatou components of the Newton maps applied
to entire transcendental functions.