$\newcommand{\RR}{\mathbb{R}}\newcommand{\QQ}{\mathbb{Q}}$One can show that there are uncountably many continuous function $f:\RR\to\RR$ with the property that $f(q)\in\QQ$ for $q\in\QQ$. It's also not hard to show that there are still uncountable many such functions which are $C^\infty$. My questions is:
Are there uncountably many analytic functions $f:\RR\to\RR$ for which $f(q)\in\QQ$ for every $q\in\QQ$.
Yes. See this beautiful answer by Makholm and note that there are uncountably many enumerations of the rationals (there are plenty of parameters open to being varied here, but I guess this is the first that catches the eye in the answer).