More accurately, let $\displaystyle A=\sum_{i=0}^{\infty}A_i$ be a finitely generated graded algebra over say $\mathbb{Q}$ but $\dim A_i=\infty$ for each $i.$ Is it possible?

$\mathbb Q[x,y]$ with $x$ in degree 0, and $y$ in degree 1. If you want your generators to be in positive degrees, then what you're asking for is impossible. 

