  We study unboundedness properties of functions belonging Lebesgue and Lorentz
spaces with variable and mixed norms using growth envelopes. Our results extend
the ones for the corresponding classical spaces in a natural way. In the case
of spaces with mixed norms it turns out that the unboundedness in the worst
direction, i.e., in the direction where $p_{i}$ is the smallest, is crucial.
More precisely, the growth envelope is given by $E_G(L_{\vec{p}}(\Omega)) =
(t^{-1/\min\{p_{1}, \ldots, p_{d} \}},\min\{p_{1}, \ldots, p_{d} \})$ for mixed
Lebesgue and $E_G(L_{\vec{p},q}(\Omega)) = (t^{-1/\min\{p_{1}, \ldots, p_{d}
\}},q)$ for mixed Lorentz spaces, respectively. For the variable Lebesgue
spaces we obtain $E_G(L_{p(\cdot)}(\Omega)) = (t^{-1/p_{-}},p_{-})$, where
$p_{-}$ is the essential infimum of $p(\cdot)$, subject to some further
assumptions. Similarly, for the variable Lorentz space it holds
$E_G(L_{p(\cdot),q}(\Omega)) = (t^{-1/p_{-}},q)$. The growth envelope is used
for Hardy-type inequalities and limiting embeddings. In particular, as a
by-product we determine the smallest classical Lebesgue (Lorentz) space which
contains a fixed mixed or variable Lebesgue (Lorentz) space, respectively.
