Description: Numerical models of mantle convection typically employ a temperature- or pressure-dependent viscous or viscoplastic rheology and a free slip upper boundary condition. The Earth, however, has a stress-free rather than a free slip surface condition. In addition, with decreasing temperature, the viscosity of rocks increases, which might induce a change from viscous to elastic behaviour (depending on the timescale of deformation). Here, we study the effects of both a Maxwell viscoelastic rheology and a free surface upper boundary condition on viscoelastic convection with a strongly temperature dependent rheology. We particularly focus on the effect of elasticity on the stress state of the lithosphere. Results show that convection vigor and heat transport are not significantly altered by the upper boundary condition or by elasticity. However, the stress state of the lithosphere is significantly affected by both factors. If elasticity is unimportant, a free surface upper boundary condition results in significantly elevated surface stresses (which are up to two magnitudes larger than in the free slip case). Elasticity counteracts this effect and significantly reduces the surface stresses, but distributes stresses over a thicker layer than in the case of a purely viscous rheology. At Earth-like conditions, this effect is significant. While it is warranted to use a free slip upper boundary condition and neglecting elasticity when studying mantle convection and its effect on the thermal state of the Earth, both factors are significant when one wants to predict the stress state of the lithosphere and related questions. Additional 2-D simulations of a plume impinging on a constant thickness and constant viscosity lithosphere show that reasonable parameters might induce lithospheric stress levels that are on the order of a GPa or larger, for viscous free surface models, and that these stresses are several orders of magnitude larger than stresses that occur for free slip models. This suggests that the effect of a free surface should not be ignored in models where the rheology is stress-dependent, such as in viscoplastic models of self-consistent plate tectonics.