I have the following question (I am not a mathematician, so I am sorry if it is ill-formed!).
Some irrational numbers such as $\pi$ are suspected to be normal in that any finite string of digits is 'equally likely' to appear in their decimal expansion. As I understand it, it is an open question whether $\pi$ is normal.
Are there any irrational numbers of which we cannot know whether they are normal, so that the only way to check whether such a number has infinitely many zeros, for example, would be to 'go through' all of its decimal expansion?
Are you familiar with the concept of countable and uncountable sets? In particular that the real numbers are uncountable? If not then see Cantor's diagonal argument.
The rational numbers are countable and hence a strict subset of the reals. There are other interesting sets which are intuitively larger yet still countable.
The algebraic numbers include $\sqrt 2$ and other numbers constructed with roots but more generally any number which is the solution to a polynomial with integer coefficients.
The computable numbers are those for which we have an algorithm to compute them to any desired precision. Note that this just means computable in a theoretical sense not any practical one. $\pi$ and $e$ are examples. We can in principle compute the googolth ($10^{100}$) digit of $\pi$. It is highly unlikely that we ever will as a googol is way more than the number of atoms in the visible universe. Although intuitively bigger again, this set is still countable hence some real numbers must not be computable.
These uncomputable numbers might be what you are looking for. There is no algorithm for their digits, they could only be specified by writing them out.
Can we give an example of any? Well sort of. We can look at an even larger set: the definable numbers. A definable number has a definition which is in a sense exact but cannot be computed. Maybe the closest to a concrete example of a definable but uncomputable number is Chaitin's Constant. This set is still countable so there are real numbers which are not even definable. If a merely non-computable number did not satisfy your requirements then maybe a non-definable one does.