Are there real numbers that can only be expressed via a complex expression?

Question

Someone recently told me certain real numbers could only be expressed in closed form via an expression involving complex numbers.

Is this true? If so do these numbers have a name?

What is a simple example?

Answer 1

This phenomenon occurs prominently in the (historically interesting) casus irreducibilis of third degree polynomial equations with three real roots. Take the equation $$x^3-2x^2-6x+5=0$$ as an example.

Answer 2

To answer my final part about a nice, minimal example:

$$\sqrt[3]{1+i \sqrt{7}}+\sqrt[3]{1-i \sqrt{7}}$$

Which $\approx$ 2.6016791318831542525

This value was adapted from one of the roots of the polynomial posted in the accepted answer, then back solved for the generating cubic to ensure it is a real casus irreducibilis. (Previous version using $\sqrt{5}$ did not have a rational cubic.)

The generating cubic is simply $x^3 - 6x - 2 = 0$.

As noted by a @Steven Stadnicki , this is only irreducible under the radicals. Using Euler's formula we can find a trigonometric representation of this number.

$$2 \sqrt{2} \cos{\left(\frac{\tan^{-1}{\left(\sqrt{7}\right)}}{3}\right)}$$