We investigate the effect of highly contrasting non-Newtonian liquid properties on the formation of liquid jets with a focused shape. By using two nozzle-free ejection techniques, mechanically impact- and laser-induced, fast jets of a highly elastic (sodium polyacrylate) and weakly elastic (xanthan-gum) diluted polymer solutions are generated. A unique high-speed effect is encountered at the jet ejection onset of the highly elastic solution. Its jet-tip speed is on average 1.4 times faster in comparison to a Newtonian (glycerin/water) and the weakly elastic liquids. We explain this effect occurring as a result of the high viscoelasticity of the sodium polyacrylate solution. Additionally, a 'bungee jumper' jet behaviour (Morrison and Harlen in Rheol Acta 49(6):619-632, 2010) is observed in a regime of high speed (10 < V-j < 40 m/s) and high viscosity (mu > 20 mPa s) not previously examined. We additionally characterise the viscoelastic non-breakup jet limit using the Bazilevskii et al. (Fluid Dyn 40(3):376-392, 2005) ejection criterion. Herein, the extensional rheological parameters are measured implementing a novel DoS-CaBER technique (Dinic et al. in Lab Chip 17(3):460-473, 2017). Our findings may influence results of inkjet printing technologies and recent nozzle-free ejection systems for ejecting liquids with non-Newtonian properties.