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Let $\Omega$ be a bounded open set in ${ℝ}^n$, $n\ge 2$. In a well-known paper , 20, 1077–1092 (1971) Moser found the smallest value of $K$ such that $$sup\left\{{\int }_{\Omega }exp\left({\left(\frac{\left|f\left(x\right)\right|}{K}\right)}^{n/(n-1)}\right):f\in W_0^{1,n}\left(\Omega \right),{\parallel \nabla f\parallel }_{L^n}\le 1\right\}<\infty .$$ We extend this result to the situation in which the underlying space $L^n$ is replaced by the generalized Zygmund space $L^n{log}^{n-1}L{log}^{\alpha }logL$ $(\alpha <n-1)$, the corresponding space of exponential growth then being given by a Young function which behaves like $exp(exp\left(t^{n/(n-1-\alpha )}\right))$ for large $t$. We also discuss the case of an embedding into triple and other multiple exponential cases.