It seems that indeed only prime fields are unrepeatable.
Proof: Let $k$ be unrepeatable and $F\subseteq k$ denote the prime field of $k$. Note that in particular then $\mathrm{Aut}(k)$ is trivial. Let $T\subseteq k$ be a transcendence base of $k/F$ and let $G=F(T)$. If $T\neq\emptyset$, then $G/F$ has non-trivial automorphisms (say take one element $t\in T$ to $-t$ (oops, this assumes $\mathrm{char} k\neq 2$, so I need to add some love to this). Since $k/G$ is algebraic this extends to a non-trivial automorphism of $k$. Therefore, we see that $k/F$ must be algebraic.
It follows that $F\subseteq k\subseteq \overline{F}$ and since $k$ is unrepeatable (in $\overline F$), $k$ has to be fixed by all automorphisms in the absolute Galois group of $F$. But then $F=k$ and we are done.