Conformal theories with a global symmetry may be studied in the double scaling regime where the interaction strength is reduced while the global charge increases. Here, we study generic 4d N = 2 SU (N ) gauge theories with conformal matter content at large R-charge Q R → ∞ with fixed 't Hooft-like coupling κ = Q R g 2 YM . Our analysis concerns two distinct classes of natural scaling functions. The first is built in terms of chiral/anti-chiral two-point functions. The second involves one-point functions of chiral operators in presence of 1 2 -BPS Wilson-Maldacena loops. In the rank-1 SU (2) case, the two-point sector has been recently shown to be captured by an auxiliary chiral random matrix model. We extend the analysis to SU (N ) theories and provide an algorithm that computes arbitrarily long perturbative expansions for all considered models, parametric in the rank. The leading and next-to-leading contributions are cross-checked by a three-loops computation in N = 1 superspace. This perturbative analysis identifies maximally non-planar Feynman diagrams as the relevant ones in the double scaling limit. In the Wilson-Maldacena sector, we obtain closed expressions for the scaling functions, valid for any rank and κ. As an application, we analyze quantitatively the large 't Hooft coupling limit κ 1 where we identify all perturbative and non-perturbative contributions. The latter are associated with heavy electric BPS states and the precise correspondence with their mass spectrum is clarified. arXiv:2001.06645v2 [hep-th] 28 Jan 20201 The idea of a large charge/weak coupling compensation is closely related to the solvability of the BMN limit in AdS/CFT [4,5] and, more generally, to the coherent-state effective theory description of "semiclassical" string states [6,7] and its role in capturing the strong coupling regime [8].2 An equivalent statement is that an exact saddle point analysis is possible in the double scaling limit.