Given an element $\displaystyle x\in \mathbb{R}$ having a property that no non-constant polynomial in $\displaystyle \mathbb{Q}$ has $\displaystyle x$ as a zero, we call it "transcendental". For example, $\displaystyle \pi \in \mathbb{R}$ is an example.

However, if we allow "infinite polynomials", i.e. power series then it is no longer transcendental. Because $\displaystyle \pi$ is a zero of $\displaystyle f(x)=x - \frac{x^3}{3!}+\frac{x^5}{5!}-...$.

My question is: Is there an element in $\displaystyle \mathbb{R}$ which is purely transcendental? Meaning it has no rational power series even.