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.
A real number should really just be the concept of a number, meant in the broadest sense. Strictly speaking, quaternions, complex numbers, elements of finite fields, etc. are not numbers.
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 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 x_i : \{0,1\}} 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 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 x_i = 1} 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. [TODO wtf does that mean]
The Euler-Mascheroni constant (where we don't actually know for sure whether or not it is rational) [TODO I'm not sure if it really makes sense]