We are interested in the nearly supercritical regime in a family of maxtype recursive models studied by Collet, Eckman, Glaser and Martin (Comm. Math. Phys. 94 (1984) 353-370) and by Derrida and Retaux (J. Stat. Phys. 156 (2014) 268-290) and prove that, under a suitable integrability assumption on the initial distribution, the free energy vanishes at the transition with an essential singularity with exponent J. This gives a weaker answer to a conjecture of Derrida and Retaux (J. Stat. Phys. 156 (2014) 268-290). Other behaviours are obtained when the integrability condition is not satisfied.