Biconical-type antennas featuring high directivity have been designed, created, and tested in anechoic chamber. Results in the range between 1 and 5 GHz are presented in this article. In particular, two different configurations have been tested, with and without dielectric lenses, both involving rapid prototyping tools (3D printing) for the dielectric and the antenna support. A very high directivity is nowadays demanded by efficient and sustainable point-to-point communications or energy transfer protocols, to avoid releasing energy in neighboring areas and preserve data transfer security. As demonstrated here, special biconical type antennas featuring a 3D printed polylactic acid (PLA) dielectric lens can achieve a good directivity, with a corresponding emission lobe centered around 8.4 degrees, featuring a FWHM of 6.4 degrees. Dielectric lens-free antennas, featuring an unconventional shape, can also achieve a good directivity, with a corresponding emission lobe centered around 10.0 degrees, featuring a FWHM of 14.2 degrees. The preliminary results shown here explore some of the aspects of the vast configuration space (which include fabrication techniques, dielectric materials, conductive supports, etc.) and open the route for further optimization studies. The aim would be to adjust the various degrees of freedom in order to achieve what can be defined as "infinite" directivity.