For $E$ a fixed elliptic curve (or more generally, abelian variety) over $\mathbb Q$, the condition "$E(K)$ is a finitely generated abelian group." It is satisfied by any finite extension $K$ of $\mathbb Q$, by the theorem of Mordell-Weil. If $K \subset L$ and the condition is satisfied by $L$, then it is also satisfied by $K$ since $E(K)$ is a subgroup of $E(L)$ and a subgroup of a finitely generated abelian group is still finitely generated.
The condition is definitely not satisfied for $K=\overline{ \mathbb Q}$. But a theorem of Ribet states that this condition is satisfied for $K$ the maximal abelian extension of $\mathbb Q$ unramified outside a fixed set of primes. For other infinite extensions of $\mathbb Q$, the situation is very complex (certainly not fully understood even conjecturally), but there are a good deal of deep conjectures, and a good lot of them proved, concerning extensions of number fields whose Galois group is a pro-$p$-group (especially of the form $\mathbb Z_p^n$). For examples of such conjectures and theorem, see Greenberg's Introduction to Iwasawa Theory for Elliptic Curves.