Bounds on size of self-concatenation of Finite Languages
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.