Compute e^x given starting approximation
The obvious answer to "I have an approximate solution to an equation and I want to improve it" is Newton-Raphson. If you have an approximation $y \approx e^x$ then you can improve it to $$y' = y - \frac{(\ln y) - x}{y^{-1}} = y(1 + x - \ln y)$$

Now you need an efficient way of calculating $\ln y$, but here I note that Wikipedia, commenting on high precision logarithms, says (my emphasis):


  Another alternative for extremely high precision calculation is the formula
  $$\ln x \approx {\frac {\pi }{2M(1,4/s)}}-m\ln 2,$$
  where $M$ denotes the arithmetic-geometric mean of $1$ and $4/s$, and
  $$\displaystyle s=x2^{m}&gt;2^{p/2},$$
  with $m$ chosen so that $p$ bits of precision is attained. (For most purposes, the value of $8$ for $m$ is sufficient.) In fact, if this method is used, Newton inversion of the natural logarithm may conversely be used to calculate the exponential function efficiently.


The references given are Practically fast multiple-precision evaluation of log(x), Sasaki and Kanada (1982), Journal of Information Processing 5(4) pp247–250; and Fast Computations of the Exponential Function, Ahrendt (1999), Stacs Lecture Notes in Computer Science 1564 pp302–312.