For example, a Turing machine might take input which comprises two numbers and then produce output which is the product of their multiplication.
More formally, it is the probability measure of reals (infinite binary sequences) which have the property that every initial segment of them preserves the universality of the given Turing machine.
It was shown that when the underlying machine is universal, these numbers are highly algorithmically random.
In other words, they are random relative to null sets that can be defined with four quantifiers in Peano arithmetic.
Universality probability provides a concrete and somewhat natural example of a highly random number (in the sense of algorithmic information theory).