In this paper, we study combinatorial and asymptotic properties of some interesting rational numbers called the Brézin--Gross--Witten (BGW) numbers, which can be represented as the intersection numbers of psi and Theta classes on the moduli space of stable algebraic curves. In particular, we discover and prove the uniform large genus leading asymptotics of certain normalized BGW numbers, and give a new proof of the polynomiality phenomenon for the large genus asymptotics. We also propose, with extensive numerical data, several new conjectures including monotonicity and integrality on the BGW numbers. Applications to the Painlevé II hierarchy and to the BGW-kappa numbers are given.