Bounds on size of self-concatenation of Finite Languages 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 $|L| &gt; 1$, but I would like to know if there is either a tight upper bound or tight lower bound.
You can show $ |L|^i $ is a tight upper bound by using the following language:


  $ L = \{ ab,aab,aaab,\ldots,a^kb \mid k \geq 1 \}. $


Any concatenation gives a new string. For a lower bound, I can suggest the following unary language:


  $ U = \{a,aa,aaa,\ldots,a^k \mid k \geq 1 \} $. 


Then, $ U^i = \{ a^i,a^{i+1},\ldots,a^{ki} \} $ and so $ |U^i| = i|U|-i+1 $.

Emil's (Jerabek) comment appeared during writing my answer and so before posting my answer.