Span of Transcendental Numbers

Question

Let $a$ and $b$ be two transcendental numbers. Does there exist $r \in \mathbb{R}$ such that $r$ cannot be expressed as any finite (integral) powers of $a$ and $b$ with rational coefficients?

For any finite $n \in \mathbb{Z}$, does the following hold?

$\sum_{i=0}^n c_ia^i + d_ib^i = r$

where $c_i, d_i \in \mathbb{Q}$

Edit: Since there are answers that $r$ exists, please provide an example given any two transcendental numbers $a$ and $b$.

That is find $r\notin\{\,\sum_{i=0}^n(c_ia^i+d_ib^i)\;|\; c_i,d_i\in \Bbb Q\land 0\le n\in \Bbb Z\}$

Answer 1

Yes. Otherwise, the transcendence degree of $\mathbb R$ over $\mathbb Q$ over be smaller than or equal to $2$. But it is actuallly infinite (in fact, it has the cardinality of the continuum).

Answer 2

$\{\,\sum_{i=0}^n(c_ia^i+d_ib^i)\;|\; c_i,d_i\in \Bbb Q\land 0\le n\in \Bbb Z\}$ is countably infinite but $\Bbb R$ is uncountably infinite.