Question: or the reverse, where a repeating expansion in a variable base to be rational?
been trying some trial and error cases without success

by variable base we mean each digit can be in a different base but any digit can occur as many times in the same base or not

Just in case anybody tries to pull a fast one no base $\sqrt 2$, $\pi$ , $e$ etc. can not have the irrational number itself as base, that would make this trivial.

Answer: 
Consider $.1141144114441144441144444\dots$, where the 1s are in base 3, and the 4s are in base 9. The expansion is nonrepeating, but it represents $${1\over3}+{1\over9}+{4\over9^2}+{1\over3\cdot9^2}+{1\over9^3}+{4\over9^4}+{4\over9^5}+\cdots$$ which is $${4\over9}+{4\over9^2}+{4\over9^3}+{4\over9^4}+{4\over9^5}+\cdots={1\over2}$$ which is rational.
This is more a comment than an answet.

If the expansion is of the form
$x=\sum\_{k=0}^{\infty} d\_k/B\_k$
where $B\_{k+1} >B\_k$
and $0\le d\_k < B\_{k+1}/B\_k$
(the standard decimal system
has $B\_k=10^k$)
then the representation is unique
if and only if
$B\_{k+1}/B\_k$
is an integer for all $k$.

In the factorial system,
with $B\_k=k!$,
the representation terminates
if and only if
the number is rational.
In this system,
$e$ has
$d\_k=1$
for all $k$,
so this transcendental number
has a repeating
representation.