Question
Does there exist a (non-degenerate) distribution that does NOT belong to any maximum domain of attraction?
That is: Does there exist any non-degenerate probability distribution function $F$ such that if $X_1,X_2,\dots \overset{\text{iid}}{\sim} F$, then there do not exist any sequences $(a_n) \subset \mathbb R_{>0}$, $(b_n) \subset \mathbb R$ such that $$ \frac{\max\{ X_1, \dots, X_n\} - b_n}{a_n} $$ converges in distribution to a non-degenerate distribution?
Note: By "degenerate distribution", I mean one that takes a certain value with probability $1$, i.e. $\mathbb P(X = c) = 1$ for some $c \in \mathbb R$. Or equivalently for a distribution function $F(x) = \begin{cases} 0, &x<c \\ 1, &x \geq c\end{cases}$ for some $c \in \mathbb R$.
Thoughts
The Fisher-Tippett-Gnedenko theorem tells us that if a distribution function $F$ belongs to the maximum domain of attraction (MDA) of any non-degenerate probability distribution, then it belongs to the MDA of a generalized extreme value distribution.
The lecture notes and books I've seen covering this topic take care to emphasize that this result only hold if $F$ belongs to the MDA of some non-degenerate $G$. However, the treatments I've seen do not then offer counterexamples of non-degenerate $F$'s that don't lie in any MDA.