Special values of $j$-invariant Let $j(\tau)$ be Klein's absolute invariant defined for $\tau \in \mathbb{H}$ by
$$j(\tau) = q^{-1} + 744 + 196884q + 21493760q^2 + 864299970q^3 + \cdots$$
with $q := e^{2\pi i\tau}$.
Are there any known special values of $j(\tau)$ for which $\textrm{Re}(\tau)$ is irrational? Wikipedia has a list of special values but in all of those cases $\textrm{Re}(\tau)$ is rational.

Since it wasn't specified that $\tau$ need to be an algebraic number, then yes, there are infinitely many special values of $j(\tau)$ where $\Re(\tau)$ is irrational. However, $\tau$ has a hypergeometric closed-form.
It is known that the equation,
$$j(\tau) = n$$
can be solved for $\tau$ in terms of the hypergeometric function. Using Method 4, we find,
$$\tau = \frac{_2F_1\big(\tfrac16,\tfrac56,1,1-\alpha\big)}{_2F_1\big(\tfrac16,\tfrac56,1,\alpha\big)}\sqrt{-1}$$
where
$$\alpha=\frac{1+\sqrt{1-\frac{1728}n}}{2}$$
For example, negating the year $n=-2017$, then,
$$\tau \approx -0.273239 + 0.6868913\,i$$ 
such that,
$$j(\tau)=-2017$$
