Summary: Arithmetic differential equations or \(\delta\)-geometry exploits analogies between derivations and \(p\)-derivations \(\delta\) arising from lifts of Frobenius to study problems in arithmetic geometry. Along the way, two main classes such functions, describable as series, arose prominently namely \(\delta\)-characters of abelian schemes and (isogeny covariant) \(\delta\)-modular forms. However, the theory of these \(\delta\)-functions is not as straightforward in ramified settings. Overconvergence was introduced in [the first author and A. Saha, Banach Cent. Publ. 94, 99–129 (2011; Zbl 1244.11059)] to account for these issues which essentially imposes growth conditions extensions of these series to a fixed level of ramification; necessary as such extensions have non-trivial fractional coefficients. In this article, we introduce a rescaling process which identifies a class of \(\delta\)-functions we call totally overconvergent, which extend all the way to the algebraic closure of ring of integers of the maximally unramified extension of \(\mathbb{Q}_p\). Applications built on these functions allow one to remove boundedness assumptions on ramification. The bulk of the article is devoted to establishing that most \(\delta\)-functions arising in practice, namely those in the applications described in [the first author, Invent. Math. 122, No. 2, 309–340 (1995; Zbl 0841.14037); J. Reine Angew. Math. 520, 95–167 (2000; Zbl 1045.11025); Arithmetic differential equations. Providence, RI: American Mathematical Society (AMS) (2005; Zbl 1088.14001)], are totally overconvergent, which essentially extends results in [Zbl 1244.11059] to unbounded ramification.