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. Cayey, 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

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