Are all fractional deriviatives/integrals of $e^x$ equal to $e^x$?

Not monstruous, but not very simple. The fractional integrals/dérivatives of the exponential function involves the Incomplete Gamma function : Page 10, section 6, in the paper "The fractionnal derivation" http://www.scribd.com/JJacquelin/documents

In fact, the fractional integals/dérivatives are based on the Riemann-Liouville operator in which the lower bound (a) of the integral plays an important role :

http://mathworld.wolfram.com/Riemann-LiouvilleOperator.html

The conventional fractionnal calculus states a=0 :

http://mathworld.wolfram.com/FractionalCalculus.html

In this case, the fractional transform of the exponential function involves the Incomplete Gamma function as already said. But in the case of a=-infinity, corresponding to the Weyl's operator, the Incomplete Gamma term fades and the exponential term remains alone. So, with the non-conventional definition based on the Weyl's operator, the transform of the exponential function is a simple exponential function.