Variational integrals at nearly linear growth appear in the theory of plasticity with logarithmic hardening, that is the borderline configuration between plasticity with power hardening and perfect plasticity. The related (very challenging) regularity theory for minima has been intensively developed over the last 25 years, see e.g. the work of Frehse & Seregin ’99, Fuchs & Mingione ’00, Bildhauer ’03, Beck & Schmidt ‘13, Beck & Bulíček & Gmeineder ’20, Di Marco & Marcellini ’20, Gmeineder & Kristensen ’22, De Filippis & Mingione ’23. In this respect, we will discuss the validity of Schauder theory for a new class of variational integrals at nearly linear growth featuring multiple phases, under optimal assumptions quantifying the rate of nonuniform ellipticity. From recent, joint work with Cristiana De Filippis (Parma), Peter Hästö (Helsinki), and Mirco Piccinini (Pisa).