Most real numbers

Most real numbers are...

• irrational : $\mathbb{Q} \subseteq \mathbb{R}$ has measure 0
• uncomputable : a number is computable $\Leftrightarrow$ exists turing machine which outputs it, but there are only countably many Turing machines
• normal (the digits of their $n$-ary expansion are uniformly distributed) : Borel's Theorem
• members of a (particular) meagre set : see Fat Cantor Sets (thanks to Jake Fiedler!)
• transcedental : Each non-transcedental number is the root of a polynomial, there are countably many polynomials, and each polynomial has finitely many roots
• S-numbers of type 1 : proven by Sprindzhuk, at least according to this book, p. 86 (I couldn't find the actual proof). Also see this Wikipedia page.
• not definable : there are only countably many formulas in ZFC
• an equivalence class of "almost homomorphisms" $f: \mathbb{Z} \to \mathbb{Z}$ where $\{f(m + n) - f(n) - f(m)\,:\,n, m \in \mathbb{Z}\}$ is finite, and $f \sim g$ whenever $\{f(n) - g(n)\,:\,n \in \mathbb{Z}\}$ is finite : see Eudoxus real number, or this more explicit paper (contribution from Nick Mahdavi)
• Martin-Löf random : given a particular constructive null cover $U_1 \supseteq U_2 \supseteq \dots$ we have $\mu\left(\bigcap_{n = 1}^\infty U_n\right) = 0$, and there are only countably many of these (as each $U_i$ is an effective open set and hence determined by a Turing machine), so the set of Martin-Löf random numbers has full measure; see Algorithmically random sequence (contribution from Jake Fiedler)
  • As it turns out, the set of all Martin-Löf random real numbers is also meagre!

Most real numbers have...

• a corresponding subset of $\mathbb{N}$ with positive upper density : most real numbers are normal, so by definition the upper density of their corresponding set is $\frac{1}{2}$
• a continued fraction expansion $a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \dots}}$ with $\lim_{n \to \infty} (a_1a_2\dots a_n)^\frac{1}{n} = K_0 \approx 2.68545$ : see Khinchin's Constant (contribution from Nick Mahdavi)
• a continued fraction expansion $a_1 + \frac{1}{a_2 + \frac{1}{a_3 + \dots}}$ with $\lim_{n \to \infty} a_n^\frac{1}{n} = e^{\pi^2/(12 \ln{2})} \approx 3.27582$ : see Lévy's constant
• an irrationality exponent of 2 : a corollary of the Borel-Cantelli lemma (also see this paper)
• effective dimension of 1 : corollary of almost every real number being Martin-Löf random; see Effective dimension