A "semi-exponential" is a function F(z) such that F(F(z))=exp(z).We show that(a) no entire-analytic semi-exponential F(z) exists;(b) no semi-exponential F(z) exists that is analytic within any interior-connected domainthat includes both the real axis, and all complex Q obeying Q=exp(Q), in its interior, and whichmaps reals→reals;(c) Analytic semiexponentials do exist that map most reals to complex numbers and which have non-analytic points;(d) We also construct a useful piecewise-analytic real→real semi-exponentialsuch that F, F', and F'' all are continuous,and F(x) is strictly increasing and strictly concave-∪, for all real x;and indeed the domain of definition of this F(z) may beslightly expanded to a long and thin complex set that includes the real axis in its interior,albeit then F becomes discontinuous at an infinite set of nonreal points.(e) But we show that no piecewise-analytic, with piece boundaries being nonemptyrectifiable differentiable curves, semi-exponentialthat maps reals→reals can be defined within any domain that includes thestrip 0≤im(z)<π.Many of our arguments may be repurposed for many other "semi" functions besides the exponential.Finally (f) we show that a real-valuedC^∞-smooth,strictly increasing, strictly concave-∪semi-exponential exists, which under certain asymptotic analyticity demands is unique.