What properties of the real numbers are almost always true and there are no (or very few) known examples of?

Question

What properties of the positive real numbers are almost always true and there are no (or very few) known examples of?

Two that come to mind are numbers that are normal in every base and numbers satisfying Khinchine's theorem (https://en.wikipedia.org/wiki/Khinchin%27s_constant).

So I wondered what other properties are like this.

Answer 1

Here's one I like: almost all numbers fail to be computable. In particular, there are only countably many finite, terminating algorithms, and consequently there are only countably many real numbers that can be expressed (to arbitrary precision) via such algorithms.

One example of a non-computable number is Chaitin's constant.

Answer 2

Almost always still allows for an infinite yet uncountable amount to get through see Cantor Set for an example on how to build such sets.

Another more trivial example would be the number is not an integer or the number is not of this "particular" countable set whatever that set may be.

Other things that come to mind are things like Zigmondys theorem which states that in the series $2^n-1$ the only number to not contain a new prime factor is $n=6, 2^6-1 =63$.