  Infinite exponential sequences of distinct prime numbers of the form $\lfloor
a c^{n^d}+b\rfloor$, $n\geq 0$, are proved to exist for well chosen real
constants $a>0$, $b$, $c>1$, $d>1$, assuming Cramer's conjecture on prime gaps.
There is an infinity of such prime sequences. Sequences having the least
possible growth rate are of particular interest. This work's focus is on prime
sequences with $a=1$, $b \in \{0,1\}$, that have the smallest possible constant
$c$ given $d>1$, and sequences with the smallest possible $d$, given $c=2$. In
particular, we prove the existence of the four infinite exponential prime
sequences $u_0(n)=\lfloor c_0^{n\sqrt{n}}\rfloor$, $n\geq 1$, with
$c_0=2.0073340803...$, $u_1(n)=1+\lfloor c_1^{n\sqrt{n}}\rfloor$, $n\geq 0$,
with $c_1=2.2679962677...$, $v_0(n)=\lfloor 2^{n^{d_0}}\rfloor$, $n\geq 1$,
with $d_0=1.5039285240...$, and $v_1(n)=1+\lfloor 2^{n^{d_1}}\rfloor$, $n\geq
0$, with $d_1=1.7355149500...$.
