Extreme values of class numbers of real quadratic fields

We improve a result of H. L. Montgomery and J. P. Weinberger by establishing the existence of infinitely many fundamental discriminants $d>0$ for which the class number of the real quadratic field $\mathbb{Q}(\sqrt{d})$ exeeds $(2e^{\gamma}+o(1)) \sqrt{d}(\log\log d)/\log d$. We believe this bound to be best possible. We also obtain upper and lower bounds of nearly the same order of magnitude, for the number of real quadratic fields with discriminant $d\leq x$ which have such an extreme class number.