Kleene closure of DFA
You asked a similar question some time ago about concatenation.  I gave the references back then and they still apply here.    See Yu, Zhuang, and K. Salomaa in their paper, "The state complexities of some basic operations on regular languages", TCS 125 (1994), 315-328, where the state blow-up for Kleene * is described in detail; it is exponential in the worst-case for languages over 2-letter alphabets or larger, with a bound of the form $3 \cdot 2^{n-2}$.  The result for Kleene * was also published (without proof) by Maslov in Dokl. Akad. Nauk SSSR (1970) (and in English in the translation journal Soviet Math. Dokl.) 

Evidently the complexity of actually computing the DFA could be exponential in the worst-case, too, then.