Growth of Mordell-Weil Rank of Elliptic Curves over Field Extensions

This is (unfortunately for you) well-known. Theorem 10.1 of this paper shows that over an algebraically closed field of characteristic zero the Mordell-Weil rank of any positive-dimensional abelian variety is infinite, and (as remark 1 below the theorem mentions) this implies that for any abelian variety over field of characteristic zero, there are finite extensions over which the ranks get arbitrarily large.

The result also holds for fields of positive characteristic which are not algebraic over a finite field.

For elliptic curves you can be more specific. For example, it's not hard to show that there are elements $a_1,\ldots,a_r\in\mathbb{F}$ so that for $\mathbb{K}=\mathbb{F}(\sqrt{a_1},\ldots,\sqrt{a_r})$ one has $\operatorname{rank}E(\mathbb{K})\ge r$. In particular, $E$ has infinite rank over the maximal abelian extension of $\mathbb F$ of exponent 2. However, if you ask for the minimal degree of an extension $\mathbb K/\mathbb F$ such that $E(\mathbb K)$ has rank $r$, then I think that you get an interesting question, although that, too, has certainly been studied.