$\pi$ and $e$ won't do it; they are eminently recursive, because we have algorithms that calculate the $n$th digit of each.
Any such number must be very weird, and (obviously) its exact numerical value must be unknown. Chaitin's constant is an example. But here is an even simpler example: Let $M$ be an enumeration of turing machines. Let $R$ be such that $R[n]$ is 1 if $M(n)$ halts on all inputs, and 0 if there is some input on which $M(n)$ does not halt. Then $R$ is clearly non recursive; if $R$ were recursive, the halting problem would be solvable. The Chaitin construction is similar. Chaitin's constant $\Omega$ is a number which in a certain sense represents the probability that a randomly-selected turing machine will halt.