In this study, we investigate the growth of bubbles within predominately extensional-deformation flows of thin film stretching form. This involves more than one free-surface to the flow (multiple surfaces), typically as inner (bubble) and outer (filament) boundaries that introduces fluid-gas interfacial treatment. Various bubble initial states and locations may be considered. The problem is discretised in space-time through a hybrid-finite element/volume pressure-correction formulation, coupled with an arbitrary Lagrangian-Eulerian (ALE) coupled with VOF scheme to track domain-mesh and free-surface movement. We contrast these results against the results from a complete ALE algorithm. Various fluid-filament materials have been considered, covering such properties as constant viscosity fluids (Newtonian), low-polymeric/high-solvent viscosity Boger-type (Oldroyd-B) fluids and high-polymeric/low-solvent viscosity elastic-type fluids (Oldroyd-B and Phan-Thien/Tanner). Numerical solutions are presented in terms of comparison between algorithms (ALE versus hybrid ALE/VOF), shapes (bubble shapes, filament shapes), contours of extra-stress (magnitude and location), mid-filament radius and extensional viscosity.