I’m seeking an explanation of how any formal language L concatenated with itself $k$ times, $L^k$, can equal that same language concatenated with itself $k+1$ times, $L^{k+1}$.

The problem I’m working on specifically asks for me to provide some language $L$ over $Σ={a}$ such that $L^3 = L^4$, but $L \neq L^2$ and $L^2 \neq L^3$. Intuitively, this does not make sense to me. Seemingly $L^{k+1}$ should always be greater in cardinality than $L^k$, and thus be unequal. I assume that the answer relates to the null string being in $L$, but I am not sure how or why.

Could anyone clarify where my intuition is leading me astray? Thank you for any insight.