A large class of recursion relations x n + 1 = λ f(xn) exhibiting infinite bifurcation is shown to possess a rich quantitative structure essentially independent of the recursion function. The functions considered all have a unique differentiable maximumbar x. Withf(bar x) - f(x) ˜ | {x - bar x} |^z (for| {x - bar x} | sufficiently small), z > 1, the universal details depend only upon z. In particular, the local structure of high-order stability sets is shown to approach universality, rescaling in successive bifurcations, asymptotically by the ratio α (α = 2.5029078750957... for z = 2). This structure is determined by a universal function g *( x), where the 2nth iterate of f, f (n), converges locally to α -n g *( α n x) for large n. For the class of f's considered, there exists a λ n such that a 2n-point stable limit cycle includingbar x exists; λ ∞ - λ n R δ -n ( δ = 4.669201609103... for z = 2). The numbers α and δ have been computationally determined for a range of z through their definitions, for a variety of f's for each z. We present a recursive mechanism that explains these results by determining g * as the fixed-point (function) of a transformation on the class of f's. At present our treatment is heuristic. In a sequel, an exact theory is formulated and specific problems of rigor isolated.