Let $v,w$ be vector fields and let $\Phi_t$ be the flow generated by $v$, that is $\frac{d}{dt}\big|_0\Phi_t(x)=v(x)$. The Lie derivative of $w$ in direction of $v$ is usually defined as $\mathcal L_vw:=\frac{d}{dt}\big|_0(\Phi_{-t})_*w\circ \Phi_t$, which is again a vector field. And $n$-fold concatenation of that operation yields $\mathcal L^n_vw$. Is it true that \begin{equation} (\Phi_{-t})_*w\circ \Phi_t=\sum_{n=0}\frac{t^n}{n!}\mathcal L^n_vw\qquad? \end{equation} The zeroth and first order terms agree, but how about the higher order terms in $t$. I strongly believe its true, but can't prove nor disprove it. I tried expanding in time and collecting terms in $t$, but this is hopeless. I feel that one should be able to proof it by induction.
Note: I consider everything to be analytic!!
The crucial observation is that:
Lemma :
\begin{equation} \frac{d}{dt}\bigg|_s (D\Phi_{-t})_{\Phi_t(x)}w(\Phi_t(x))= (D\Phi_{-s})_{\Phi_s(x)}\mathcal L_vw(\Phi_s(x)) \end{equation} By repeated application of this Lemma we can show that \begin{align} \frac{d^{k+1}}{dt^{k+1}}\bigg|_0(D\Phi_{-t})_{\Phi_t(x)} w(\Phi_t(x))=& \;\frac{d}{dt}\bigg|_{0}\frac{d^{k}}{ds^{k}}\bigg|_{t}(D\Phi_{-s})_{\Phi_s(x)} w(\Phi_s(x))\\ =&\;\frac{d}{dt}\bigg|_{0}(D\Phi_{-t})_{\Phi_t(x)}\mathcal L^k_vw(\Phi_t(x))\\ =&\; \mathcal L_v^{k+1}w(x) \end{align} So the Taylor expansion of $(D\Phi_{-\varepsilon})_{\Phi_\varepsilon(x)} w(\Phi_\varepsilon(x))$ is $w(x)+\varepsilon \mathcal L_vw+\frac{\varepsilon^2}{2!}\mathcal L^2_vw+\dots$.
Proof of Lemma: \begin{align} \frac{d}{dt}\bigg|_s(D\Phi_{-t})_{\Phi_t(x)} w(\Phi_t(x))=& \frac{d}{d\varepsilon}\bigg|_0 (D\Phi_{-s-\varepsilon })_{\Phi_{s+\varepsilon}(x)} w(\Phi_{s+\varepsilon}(x))\\ =&\; \frac{d}{d\varepsilon}\bigg|_0 (D\Phi_{-s})_{\Phi_{s}(x)}(D\Phi_{-\varepsilon})_{\Phi_{s+\varepsilon}(x)} w(\Phi_{s+\varepsilon}(x))\\ =&\; (D\Phi_{-s})_{\Phi_{s}(x)} \frac{d}{d\varepsilon}\bigg|_0 (D\Phi_{-\varepsilon})_{\Phi_{\varepsilon}(\Phi_s(x))} w(\Phi_{\varepsilon}(\Phi_s(x)))\\ =&\; (D\Phi_{-s})_{\Phi_{s}(x)}\mathcal L_vw(\Phi_s(x)) \end{align}