Real number
I am not yet sure how to define real numbers. Many irrational numbers (e.g. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \pi} ) are in fact valid concepts, but the standard definition of the reals involves infinite nonsense.
The concept of real numbers is a concept: it identifies things out in reality. In particular, it does not need to be "constructed" via some set-theoretic method like Dedekind cuts, nor do such constructions even make sense. That any real number can be approximated by fractions is obvious: a fraction is the outcome of directly measuring any quantity with a standard ruler.
Examples
Any fraction.
Any algebraic number, like Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{2}} .
The number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.x_1 x_2 x_3 x_4 \cdots} , where is its Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} th digit in base 2, and where if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} is prime, and 0 otherwise.
The Euler-Mascheroni constant (where we don't actually know for sure whether or not it is rational)