Vector Field Exponential Map I've got ${\bf v} = x^2\partial_x$, and I'm trying to find $\exp(\varepsilon{\bf v})$, but I'm having some trouble.
If I define ${\bf v}^{n+1} = {\bf v}{\bf v}^n$ then I get a different outcome to ${\bf v}^{n+1} = {\bf v}^n{\bf v}$.
For example:
$${\bf v}^2 = {\bf vv} = (x^2\partial_x)(x^2\partial_x) = x^2(\partial_xx^2)\partial_x = 
x^2(2x)\partial_x = 2x^3\partial_x$$
$${\bf v}^3 = {\bf v}{\bf v}^2 =(x^2\partial_x)(2x^3\partial_x)=x^2(\partial_x2x^3)\partial_x = x^2(6x^2)\partial_x = 6x^4\partial_x$$
$${\bf v}^3 = {\bf v}^2{\bf v} = (2x^3\partial_x)(x^2\partial_x) = 2x^3(\partial_xx^2)\partial_x = 2x^3(2x)\partial_x = 4x^4\partial_x$$
Using ${\bf v}^{n+1} = {\bf v}{\bf v}^n$ gives 
$$\exp(\varepsilon {\bf v})x = \frac{x}{1-\varepsilon x}$$
While using ${\bf v}^{n+1} = {\bf v}^n{\bf v}$ gives
$$\exp(\varepsilon {\bf v})x = \frac{x}{2}(1+\mathrm e^{2\varepsilon x})$$
In both cases, when $\varepsilon =0$, we get just $x$, i.e. the identity element. Also, in both cases, we get
$$\lim_{\varepsilon \to 0} \frac{\mathrm d}{\mathrm d\varepsilon} \exp(\varepsilon {\bf v})x = {\bf v}$$
The same ${\bf v} \in \mathfrak g$ can't possible generate two different flows, can it?

You are looking at the simplest Lie advective flow in perturbation theory (as applied in physics: QFT) and the workhorse example in the 19th century book of Georg Sheffer cited.
v generates a shift operator, and it pays to define suitable canonical coordinates,
$$
y=-1/x, \qquad \Longrightarrow \qquad x^2 \partial_x=\partial_y ,
$$ 
so that you are shifting y by $\epsilon$,
$$
e^{\epsilon \partial_y} ~~f(y)= f(y+\epsilon),
$$
which reads
$$
e^{\epsilon x^2\partial_x} ~~g(x)= g\left(\frac{-1}{y+\epsilon}\right )=g\left (\frac{x}{1-\epsilon x}\right ), 
$$
a standard formula in the RG advection of QFT.
Your difficulties are traceable to your refusal to perform your Heaviside calculus manipulations of functions of differential operators with a "test-function" f(x) on the right, which keeps track of the proper chain rule action of noncommuting derivative operators.
