No, there is not (well, "$0=1$" ... :P). You're quite right that from a first-order definition of Reinhardt-ness, we would conclude immediately the nonexistence of Reinhardt cardinals. Indeed, more can be said: Suzuki proved (doi: 10.2307/2586799)that there is no definable-from-parameters element...

For example, the completeness of real numbers was discovered around 1800 and now $\mathbb{R}$ is considered to be a Dedekind-complete ordered field. Is it possible that a new axiom about $\mathbb{R}$ will be discovered?

While the OP specifically asks about $\mathbb{R}$, I'm going to say a bit about the more general problem of additional axioms for mathematics in general. The case of $\mathbb{R}$ plays a special role here of course, but I think it helps to view it in the broader context. Because this answer is r...