Classes of languages for which the language $\{a_1^n \cdots a_k^n \mid n \geq 0\}$ is in the language for $k$ but not $k+1$
One example would be the hierarchy defined by blind counter automata.
A blind $k$-counter automaton has access to $k$ counters for which


there is an increment and a decrement instruction (and no zero-test),
a run is accepting if in the end, all counters are zero, and
during a run, the counters are allowed to be negative.


Let $\mathcal{B}_k$ denote the class of languages accepted by blind
$k$-counter automata. Then $L_{k}\in \mathcal{B}_{k-1}$: while you
read $a_1$'s, you increment all $k-1$ counters simultaneously; then each of the remaining $k-1$ letter blocks decreases one of the counters.

On the other hand, one has $L_{k}\notin\mathcal{B}_{k-2}$.
This can be shown, for example, using Theorem 6.4 in Fernau and
Stiebe's paper on valence
automata/grammars. This pumping lemma allows, in a blind $\ell$-counter automaton, the simultaneous repetition of $\le\ell+1$
non-empty factors. Hence, if we had $L_k\in\mathcal{B}_{k-2}$, we could repeat $\le k-1$ factors, which can span at most $k-1$ distinct letters 
in a word $a_1^*\cdots a_k^*$. Hence, we could pump up powers of some $a_i$'s
without touching at least one letter $a_j$, which clearly leads out of
the language. In Fernau and Stiebe's paper, the class $\mathcal{B}_k$ is
denoted $\mathscr{L}(\mathrm{Val},\mathrm{Reg},\mathbb{Z}^k)$.