lhf already brought up Lagrangian inversion (a special case of the more general Lagrange-Bürmann series):
$f^{(-1)}(x)=\sum_{k=0}^\infty \frac{x^{k+1}}{(k+1)!} \left(\left.\frac{\mathrm d^k}{\mathrm dt^k}\left(\frac{t}{f(t)}\right)^{k+1}\right|_{t=0}\right)$
There are a number of nice series reversion algorithms that make use of coefficients from the series to be inverted. Here is a Mathematica implementation of an algorithm due to Henry Thacher (also used here):
a = Rest[CoefficientList[Series[(x + Sin[x])/2, {x, 0, 20}], x]]; n = Length[a]; Do[ Do[ c[i, j + 1] = Sum[c[k, 1]c[i - k, j], {k, 1, i - j}]; , {j, i - 1, 1, -1}]; c[i, 1] = Boole[i == 1] - Sum[a[[j]] c[i, j], {j, 2, i}] , {i, n}]; Table[c[i, 1]/2^i, {i, n}] {1/2, 0, 1/96, 0, 1/1920, 0, 43/1290240, 0, 223/92897280, 0, 60623/326998425600, 0, 764783/51011754393600, 0, 107351407/85699747381248000, 0, 2499928867/23310331287699456000, 0, 596767688063/63777066403145711616000}
Thacher's method expects the series to be inverted to take the form $x+\cdots$ (and the series for $x+\sin\,x$ starts out $2x+\cdots$), so some rescaling is necessary before feeding the series coefficients to the algorithm; the division by powers of $2$ at the end recovers the coefficients of the original function to be inverted.
There a lot more methods (e.g. Carleman matrices, Bell polynomials); search around for more information on them.