The derivative of the exponential map is given by (wiki):
$$ \frac{d}{dt} e^{X(t)} = e^{X(t)} \frac{1 - e^{-ad_{X(t)}}}{ad_{X(t)}} \frac{d}{dt}X(t) $$
Is there a reasonable formula for higher order derivatives: $$ \frac{d^n}{dt^n} e^{X(t)} = ?? $$
Or more ideally: $$ \frac{d}{dt_1}\dots\frac{d}{dt_n} e^{X(t_1, \dots, t_n)} = ?? $$
I have tried the direct formal argument from wiki for $t_1, t_2$ but got lost in it pretty quickly.