  We study unboundedness properties of functions belonging to generalised
Morrey spaces ${\mathcal M}_{\varphi,p}({\mathbb R}^d)$ and generalised
Besov-Morrey spaces ${\mathcal N}^{s}_{\varphi,p,q}({\mathbb R}^d)$ by means of
growth envelopes. For the generalised Morrey spaces we arrive at the same three
possible cases as for classical Morrey spaces $\mathcal{M}_{u,p}({\mathbb
R}^d)$, i.e., boundedness, the $L_p$-behaviour or the proper Morrey behaviour
for $p<u$, but now those cases are characterised in terms of the limit of
$\varphi(t)$ and $t^{-d/p} \varphi(t)$ as $t \to 0^+$ and $t\to\infty$,
respectively. For the generalised Besov-Morrey spaces the limit of $t^{-d/p}
\varphi(t)$ as $t \to 0^+$ also plays a r\^ole and, once more, we are able to
extend to this generalised spaces the known results for classical Besov-Morrey
spaces, although some cases are not completely solved. In this context we can
completely characterise the situation when ${\mathcal
N}^{s}_{\varphi,p,q}({\mathbb R}^d)$ consists of essentially bounded functions
only, and when it contains regular distributions only.
