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We prove existence of scaling limits of sequences of functions defined by the recursion relation w′n+1(x) = –wn(x)2. which is a successive approximation to w′(x) = –w(x)2, a simplest non-linear ordinary differential equation whose solutions have moving singularities. Namely, the sequence approaches the exact solution as n → ∞ in an asymptotically conformal way, wn(x) $\\asymp$ qn $\\bar w$(qnx), for a sequence of numbers {qn} and a function $\\bar w$. We also discuss implication of the results in terms of random sequential bisections of a rod.