A nice example.  Goes back to Cayley, 1860 [1].

Formal power series of the form
$$
f(x) = x + a_1x^2+a_2x^3+a_3x^4+\cdots
$$
with real coefficients.  Under composition.  Cayley showed how to do "fractional" composites, of real (or even complex) order.  The series need not converge, even if $f$ does.  That's why I said "formal" power series.  

**plug:** I did a generalization for transseries [2].

[1] A. Cayley, On some numerical expansions.  *Quarterly Journal of Pure and Applied Mathematics* **3** (1860) 366--369

[2] G. Edgar, Fractional iteration of series and transseries.  *Transactions of the American Mathematical Society* **365** (2013) 5805--5832