TITLE: Hard to compute real numbers
QUESTION [10 upvotes]: Turing proved that not all real numbers are effectively computable. In the sense that no algorithm exists to compute some real numbers. 
Here is Turing's definition: A real number is computable if its digit sequence can be produced by some algorithm or Turing machine. The algorithm takes an integer $ n \geq 1$ as input and produces the  $n$-th digit of the real number's decimal expansion as output.
I am interested in what is known about the efficiency of representing real numbers and decoding speed. Specifically, I want to know the relationship between the compactness of the algorithm (that encodes a real number) and the speed of producing the digits of a real number (decoding speed as a function of $n$). A real number is efficiently computable if the number of steps needed to compute digit $n$ is bounded by a polynomial in $n$. It is hard to compute if no such polynomial bound exists.

Are there any real numbers that are hard to compute (not necessarily computable one)? What are the references?

I have no formal definitions in mind.

REPLY [6 votes]: Noah Schweber has answered the question, but here is a simple example that may help drive the point home more intuitively.

List all graphs.  Let $r$ the be real number whose $n$th binary digit is 1 if and only if the $n$th graph is Hamiltonian.

In other words, any computational problem with a yes/no answer can be encoded in the binary expansion of a real number.  Since there exist hard computational problems, there exist hard-to-compute real numbers.