It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$.
The picture is less clear if $K$ is infinite-dimensional over $\mathbb{Q}$.

I believe the best result is that of Kobayashi who proved (modulo the usual conjectures on Hasse-Weil $L$-functions and BSD) that $\operatorname{rank}(E(\mathbb{Q}^{\operatorname{ab}}))=\infty$.

What is known about the rank of $E$ over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5},...)$?

EDIT: I have a follow-up question that is more precise:
http://mathoverflow.net/questions/244230/is-there-an-infinite-family-of-primes-q-1-q-2-so-that-the-rank-of-e