Given a finite language $L$ with $|L|$ number of elements, what is $|L^i|$ (the language $L$ concatenated with itself $i$ times)? If there is no exact result, is there an upper/lower bound?
Define concatenation of $L$ with itself to be as the classical definition: $LL = \{wx \space | \space w,x \in L\}$. Similarly, $L^n = \{a_1a_2...a_n \space | \space a_i \in L, \forall i \in [1, n] \}$.
Clearly, a non-tight upper bound is $|L|^i$ (each element in concatenation is unique) given that $|L| > 1$, but I would like to know if there is either a tight upper bound or tight lower bound.