What is the minimal extension of FO that captures the class of regular languages?
FO(LFP) captures PTIME on ordered structures, and strings are ordered structures.  So the languages definable by FO(LFP) include all regular languages and much much more.
http://dx.doi.org/10.1016/S0019-9958(86)80029-8

Ebbinghaus and Flum's textbook contains an exercise that asks to show FO(TC^1) (first-order logic extended with transitive closures of binary relations) is equivalent to MSO on strings. In the same exercise, $\{ a^n b^n \mid n \geq 1 \}$ is used as an example to show FO(TC^2) is more expressive than MSO on strings.  All FO(TC) formulas are clearly equivalent to FO(LFP) formulas.